Describing Function Technique Research Articles

Auto-tuning method for PI controllers under Symmetric-Send-on-Delta sampling strategy

In this work, an auto-tuning method for PI controllers applied in a loop under a Symmetric-Send-On-Delta sampling strategy is proposed. Auto-tuning algorithms usually implement an identification phase to obtain some information about the system dynamic that is used for tuning the controller. In our proposal, two frequency response points are estimated through an iterative procedure. With this information, a simple tuning rule is applied which guarantees robustness against limit cycle oscillations that can be induced by the SSOD sampler. Both the robustness measure to limit cycles and the identification procedure are based on the describing function technique, whose validity in the networked control systems under study requires a low incidence of losing packets as well as the transmission delays to be known or negligible compared to the system’s dynamic. Comparatives with other classical tuning rules support the suitability of the proposed method. The auto-tuning procedure has been validated through a simulation study reflecting its applicability for the most common dynamics found in industrial processes.

Study of the feasibility regions for a kind of event-based PI controllers

In this work we analyze the stability regions for PI controllers under a Symmetric-Send-On-Delta sampling strategy. This event-based sampling provides a significant reduction in the data needed for control and certain immunity against noisy signals without affecting significantly the performance. Hitherto, most of the analysis of this kind of event-triggered strategies to determine the apparition of limit cycles are based on the Describing Function technique, which may provide misleading results. Instead, in this work the analysis is based on the Tsypkin’s margin, which provides accurate results. Through this robustness measure, the stability regions for PI controllers are obtained as a function of its parameters. Some well-known tuning rules are also evaluated to determine its stability.

Nonlinear Active Disturbance Rejection Control Strategy for Permanent Magnet Synchronous Motor Drives

Active disturbance rejection controller (ADRC) possesses the considerable capability of improving the robustness of the permanent magnet synchronous motor (PMSM) drives. Nevertheless, the traditional linear ADRC (LADRC) and nonlinear ADRC (NLADRC) have their specific characteristic disturbance range of application. To further expand the disturbance range so as to improve the robustness against the disturbances, a novel NLADRC is proposed, which replaces the traditional linear function with a nonlinear function (NLF). And it is applied as the speed regulator for PMSM drives. Considering the accuracy and the dynamic performance of the disturbance estimation at the same time, an NLF is developed, and its describing function is established accordingly. Based on the above analysis, the NLADRC employing a cascaded nonlinear extended state observer is constructed to guarantee the relatively rapid and accurate disturbances estimation and compensation. Then, via the describing function technique, the theoretical frequency-domain analysis for the stability, the disturbance estimation performance, and the stiffness of the proposed NLADRC-based sensorless PMSM drive is given. Finally, the validity of the proposed scheme is verified experimentally on a 2.2-kW PMSM drive platform.

Improving the identification from relay feedback experiments

This paper proposes a new method for systems identification from relay feedback test. As a difference with most of the existing methods, which are based on the describing function technique, our proposal considers a more precise approach that allows to improve the estimation results. Furthermore, the method has also the advantage of being easy to implement, which makes it to stand out among other methods that require introducing additional elements on the loop or extend the experiment duration. The validity of the proposed method has been proved using a batch of models that represent the most common dynamic behaviors of actual industrial processes. The study reveals that with the new proposal, using data from a simple relay experiment, the error in the estimation for some kind of systems can be considerably reduced with respect to previous approaches.

A Novel Ultrafast Transient Constant on-Time Buck Converter for Multiphase Operation

Constant on-time control (COT) has been widely adopted in buck converters because of its simplicity, fast transient response, and high light-load efficiency. However, the increasing power demands of CPUs and AI/ML engines is pushing voltage regulators to support higher output currents, which typically requires the use of multiphase operation. However, since most high-performance COT schemes utilize ripple as a control variable, the ripple cancellation point in multiphase designs imposes difficult stability concerns, limiting performance, and robustness. In this article, a novel ultrafast transient COT (UFTCOT) control scheme is proposed to enable fast and robust operation even in multiphase converters. The small-signal models of the proposed control scheme are derived based on a describing function technique, whose theoretical results are compared, along with simulation and measurement results, to contrast the proposed UFTCOT technique to exiting approaches.

Event-based discrete PI controllers robustness analysis through sampled describing function technique

ABSTRACT In this work, the robustness of discrete PI controllers when used with a Symmetric-Send-On-Delta (SSOD) sampling law is addressed. Hitherto, the continuous Describing Function has been employed as a suitable tool to evaluate the robustness of such systems to limit cycle oscillations induced by the SSOD. However, due to the discrete implementation of the controllers in most of the actual applications, the Sampled Describing Function technique is used in this work to provide a more realistic approach, which takes into account the effect of both the SSOD and sampling period of the discrete controller on inducing such oscillation. A simple measure has been developed to characterise the robustness of these systems and it has been tested through several examples, showing its validity in predicting the apparition or avoidance of limit cycles. This measure has been used to evaluate the robustness of some spread tuning rules applied to a wide batch of systems reflecting the dynamics of most processes in the industry.

Dynamics of Cardiovascular Muscle Using a Non-Linear Symmetric Oscillator

In this paper, a complete non-linear symmetric oscillator model using the Hamiltonian approach has been developed and used to describe the cardiovascular conduction process’s dynamics, as the signal generated from the cardiovascular muscle is non-deterministic and random. Electrocardiogram (ECG) signal is a significant factor in the cardiovascular system as most of the medical diagnoses can be well understood by observing the ECG signal’s amplitude. A non-linear cardiovascular muscle model has been proposed in this study, where a modified vanderPol symmetric oscillator-based equation is used. Gone are the days whena non-linear system had been designed using the describing function technique. It is better to design a non-linear model using the Hamiltonian dynamical equation for its high accuracy and flexibility. Varying a non-linear spring constant using this type of approach is more comfortable than the traditional describing function technique. Not only that but different initial conditions can also be taken for experimental purposes. It never affects the overall modeling. The Hamiltonian approach provides the energy of an asymmetric oscillatory system of that cardiovascular conduction system. A non-linear symmetric oscillator was initially depicted by the non-linear mass-spring (two degrees of freedom) model. The motion of an uncertain non-linear cardiovascular system has been solved considering second-order approximation, which also demonstrates the possibility of introducing spatial dimensions. Finally, the model’s natural frequency expression has also been simulated and is composed of the previously published result.

A MIXED TIME-FREQUENCY DOMAIN APPROACH FOR THE QUALITATIVE ANALYSIS OF AN HYSTERETIC OSCILLATOR

Frequency domain techniques, like harmonic balance and describing function, are classical methods for studying and designing electronic oscillators and nonlinear microwave circuits. In most applications spectral techniques have been used for determining the steady-state behavior of nonlinear circuits that exhibit a single periodic attractor. On the other hand, the global dynamics of nonlinear networks and systems is usually investigated through time-domain techniques, that require to introduce rather complex and sophisticated concepts. Recently some HB based techniques have been proposed for investigating bifurcation processes in nonlinear circuits that present several attractors (the authors have considered systems that admits of a Lur’e representation). Their approach presents the advantages of providing a simple and qualitative description of the system dynamics, that can be effectively exploited for design purposes. In this manuscript we will examine a third order hysteretic oscillator, that cannot be decribed in the classical Lur’e form and we will show that its dynamics can be investigated through the joint application of the describing function technique and of a suitable time-domain method for estimating Floquet’s multipliers.

A New Approach for Nonlinear Transformation of Means and Covariances in Direct Statistical Analysis of Nonlinear Systems

Covariance analysis describing function technique is a conventional method to solve the performance analysis of the nonlinear missile guidance system. Aiming at the faultiness of covariance analysis describing function technique and its improved method, this paper proposes a new method to solve the mean and covariance transfer issue by using the high-degree spherical-radial cubature rule. In terms of simulation models, the nonlinear missile guidance system contains the saturation link to limit the lateral acceleration command and the initial heading error is introduced. In addition, the zero-mean target evasive maneuver with exponentially-decaying autocorrelation function is chosen as the input of the missile guidance system. Furthermore, compared with the existing methods, the proposed method has higher computational efficiency, accuracy and better applicability in high-dimensional nonlinear systems by theoretical derivation and simulation results from the mean and covariance values of relative lateral separation. Finally, the presented analysis method can be used as a general method for the design and statistical analysis of multidimensional nonlinear systems.

NARMAX Identification Based Closed-Loop Control of Flow Separation over NACA 0015 Airfoil

A closed-loop control algorithm for the reduction of turbulent flow separation over NACA 0015 airfoil equipped with leading-edge synthetic jet actuators (SJAs) is presented. A system identification approach based on Nonlinear Auto-Regressive Moving Average with eXogenous inputs (NARMAX) technique was used to predict nonlinear dynamics of the fluid flow and for the design of the controller system. Numerical simulations based on URANS equations are performed at Reynolds number of 106 for various airfoil incidences with and without closed-loop control. The NARMAX model for flow over an airfoil is based on the static pressure data, and the synthetic jet actuator is developed using an incompressible flow model. The corresponding NARMAX identification model developed for the pressure data is nonlinear; therefore, the describing function technique is used to linearize the system within its frequency range. Low-pass filtering is used to obtain quasi-linear state values, which assist in the application of linear control techniques. The reference signal signifies the condition of a fully re-attached flow, and it is determined based on the linearization of the original signal during open-loop control. The controller design follows the standard proportional-integral (PI) technique for the single-input single-output system. The resulting closed-loop response tracks the reference value and leads to significant improvements in the transient response over the open-loop system. The NARMAX controller enhances the lift coefficient from 0.787 for the uncontrolled case to 1.315 for the controlled case with an increase of 67.1%.

Investigating limit cycle in a free piston Stirling engine using describing function technique and genetic algorithm

This paper focuses on evaluating a free piston Stirling engine (FPSE) using describing function method along with a genetic algorithm (GA). In this regard, first, the nonlinear terms of the free piston Stirling engine are investigated by describing function technique and then, the stability of system oscillation is demonstrated using the proposed novel method. Indeed, the mentioned idea is characterized via a graphical representation incorporating genetic algorithm. Afterwards, the free piston Stirling engine parameters including the amplitudes of power and displacer pistons, operating frequency, and phase difference between the pistons motions are estimated via this novel strategy. In the sequel, the generating work and output power of FPSE are calculated taking into account the acquired parameters. Subsequently, the simulation results are verified using the practical data of a prototype engine (i.e. SUTech-SR-1). It is found that the simulation outcomes are in a good agreement with the experimental results, through which the validity of proposed method is affirmed. Indeed, this technique not only assures the stable oscillation (i.e. the sufficient condition) of the free piston Stirling engines but also can predict the engine performance including the generated work and output power as well as the values of design parameters. Furthermore, this technique can easily be applied to design problem of other new converters such as thermos-acoustic engines as well.

Robustness of event-based discrete PI controllers. A sampled describing function approach

In this paper we present a study about the robustness of event-based discrete PI controllers. Our approach is based on Sampled Describing Function technique, which has been used to characterize the non-linear effect of the Symmetric Send on Delta (SSOD) sampling strategy on the control loop. Through several examples this technique is proven to produce accurate results in predicting limit cycle oscillations and to provide guidelines to avoid them, either by detuning the controller or increasing the sampling period.

Non Linear Control System Based Modeling of Cardiac Muscle using Describing Function and Lyapnuov Stability.

Main focus of this research work is aims towards the nonlinear analysis of human cardiac muscle by describing function technique and finding stability using lyapnuov stability theory. The nature of cardiac muscle can be modeled by mass, spring with a damper where for simplicity spring and damper are considered a linear element. In reality, it has been observed that, the characteristics of spring and damper are not linear rather nonlinear. Not only that, transportation delay (non-zero reaction time) or lag phase of cardiac muscle plays an important role to make the overall model nonlinear. The range of transportation delay for which the system is stable has been calculated here to ensure the presence of dead zone type nonlinearity in cardiac muscle. In this paper a nonlinear characteristics of the model has been analyzed considering dead zone combined with saturation. The describing function technique is used here to represent the nonlinearity. A converging stable limit cycle has been found after the analysis. Finally, lyapnuov stability theorem is applied on our proposed model and it has been that the system is asymptotically stable in the sense of lyapnuov.

Dynamical behavior of integro-differential boundary value problem arising in nano-structures via Cellular Nanoscale Network approach

Piezoelectrical solid with nano-holes under time-harmonic anti-plane load is studied in the paper. The model is defined by the system of two partial differential equations and the boundary conditions which define the generalized stress. We reduce the model into an integro-differential boundary value problem and construct Cellular Nanoscale Network (CNN) architecture. Dynamical behavior of the obtained CNN model is studied by means of describing function technique. The simulations are provided which illustrate the theoretical results. The results of this paper are applicable in the field of non-destructive testing and fracture mechanics of multi-functional materials and structural elements based on them.

Fractional-order modeling of a diode

This paper adopts the fractional calculus to model a pn semiconductor diode under sinusoidal operation. The electrical impedance spectroscopy and the describing function techniques are used for measuring the device impedance. The experimental data is approximated by means of fractional-order models. The results demonstrate that the proposed approach describes the diode impedance using a limited number of parameters, while highlighting relevant dynamic characteristics that are overlooked by classical models.

A new direct statistical analysis of nonlinear systems: CAUTDET

This paper investigates the nonlinear system analysis problem for achieving higher computational efficiency and accuracy. A novel and efficient technique, covariance analysis unscented transform describing function technique (CAUTDET), is proposed. This method combines linear system covariance analysis algorithm with describing function method based on unscented transform. Furthermore, the quasi-linearization of the nonlinear system are derived to approximate the nonlinearity, while there is no requirement of the probability density function of the nonlinear system input and needs only one time calculation. Moreover, the proposed technique can be viewed as a generalization method of the nonlinear system design and analysis. Finally, simulations are presented to illustrate the effectiveness of the presented analysis method.

Multiple input describing function analysis of non-classical aileron buzz

This paper focuses on the computational study of nonlinear effects of unsteady aerodynamics for non-classical aileron buzz. It aims at a comprehensive investigation of the aileron buzz phenomenon under varying flow parameters using the describing function technique with multiple inputs. The limit cycle oscillatory behavior of an asymmetrical airfoil is studied initially using a CFD-based numerical model and direct time marching. Sharp increases in limit cycle amplitude for varying Mach numbers and angles of attack are investigated. An aerodynamic describing function is developed in order to estimate the variation of limit cycle amplitude and frequency with Mach number and angle of attack directly, without time marching. The describing function results are compared to the amplitudes and frequencies predicted by the CFD calculations for validation purposes. Furthermore, a limited sensitivity analysis is presented to demonstrate the potential of the approach for aeroelastic design.

The three dimensional guidance system design of terminal attack phase for exo-atmospheric kill vehicle

A novel terminal guidance system with a pulse-width-pulse-frequency (PWPF) modulator based on variable dead zone is designed for exo-atmospheric kill vehicle (EKV). At first, a three dimensional (3D) missile-target engagement kinematics is established on the line-of-sight (LOS) coordinate system, the guidance law is compensated by estimated maneuvering acceleration of target from nonlinear disturbance observer (DO). Secondly, a 3D terminal sliding mode guidance law with gradually increasing robustness is proposed for EKV, and the finite time stability of closed system is strictly proven. A PWPF modulator with variable dead zone is proposed to resist the sensitivity of LOS angle with relative distance decreasing, and the stability of modulator is proven by describing function technique. Finally, the guidance performances are assessed through a numerical simulation, the proposed guidance law is better attack precision than traditional guidance law, and the PWPF modulator with variable dead zone is more reasonable in fuel allocation and smaller on miss distance than traditional PWPF modulator.

Subharmonic Instability Limits for <inline-formula><tex-math>${\rm V}^2$</tex-math></inline-formula> -Controlled Buck Converter With Outer Loop Closed/Open

General closed-form subharmonic oscillation conditions are obtained for ${\rm V}^2$ -controlled buck converters. Both constant switching period and constant on-time operations are analyzed with the outer voltage loop closed or open. Once an arbitrary linear feedback is given, the associated closed-form stability condition of the converter is readily obtained. Past research results based on the describing function technique and the Floquet theory become special cases. This paper provides an alternative and easier method to analyze subharmonic instability.

Analysis and Design of LLC Resonant Converters With Capacitor–Diode Clamp Current Limiting

This paper presents a design methodology for LLC resonant converters with capacitor-diode clamp for current limiting in overload conditions. A new fundamental harmonic approximation-based equivalent circuit model is obtained through the application of describing function techniques, by examining the fundamental behavior of the capacitor-diode clamp. An iterative procedure to determine the conduction point of the diode clamp is also given. The behavior of this type of converter is analyzed and guidelines for designing the current limiting characteristics are discussed. The characterization of a 90 W converter design using the proposed methodology is presented. The converter voltage gain and the voltage-current characteristics under different overload conditions and operating frequencies are predicted using the proposed model, which accuracies are validated against the prototype with good correlation.