Describing Function Research Articles

Determination of Self-Oscillations in Relay Control Systems

Abstract Some dynamical linear elements including on-off elements such as electro-mechanical relays and electronic components used in switching mode for some feedback control systems can present special features such as the capability to autonomously generate stable self-oscillations. This paper concerns the comparison of two approaches enabling to determine the frequency of self-oscillations in these systems. We examine Tsypkin’s approach which can provide analytical solutions to determining the frequency of existing self-oscillations. On the other hand, we examine the Describing Function (DF) approach which has been developed as an alternative to approximate these solutions. We will compare these two approaches for nonlinear systems of first and second orders. We will examine particularly the possibilities and limits of each approach for calculating the self-oscillations. Simulation of these systems will be performed to visualizes their behaviors. An experimental feedback control system based on electronic circuits used in switching mode has been built as a setup enabling testing and envisioning some applications.

Stability analysis of dynamic nonlinear interval type-2 TSK fuzzy control systems based on describing function

This paper focuses on the limit cycles prediction problem to discuss the stability analysis of dynamic nonlinear interval type-2 Takagi–Sugeno–Kang fuzzy control systems (NIT2 TSK FCSs) with adjustable parameters. First, in order to alleviate computational burden, a simple architecture of NIT2 TSK FCS using two embedded nonlinear type-1 TSK fuzzy control systems (NT1 TSK FCSs) is proposed. Then, describing function (DF) of NIT2 TSK FCS is obtained based on the DFs of embedded NT1 TSK FCSs. Subsequently, integrating the stability equation and parameter plane approaches provides a solution to identify the limit cycle and the asymptotically stable regions. Moreover, particle swarm optimization technique is applied to minimize the limit cycle region. Furthermore, for robust design, a gain-phase margin tester is utilized to specify the minimum gain margin ( $$\hbox {GM}_{\mathrm{min}}$$ ) and phase margin ( $$\hbox {PM}_{\mathrm{min}}$$ ) when limit cycles can arise. Finally, two simulation examples are considered to validate the advantages of the presented method.

Validating Continuous Tuning Rules for Event-Based PI Control of Lag-Dominant Processes

One of the difficulties of the tuning of event-based proportional-integral (PI) controllers using symmetric send-on-delta sampling (SSOD) is the appearance of a stable limit cycle, especially when rules designed for continuous control loops are applied. This oscillation is explained by the intersection in the Nyquist map of the system, that is, of the loop transfer function, with the negative reciprocal of the describing function (DF) of the SSOD sampler. However, as the DF theory is based on neglecting the high-order harmonics in the closed loop system, it introduces errors in the prediction of the oscillations. The paper presents an experimental study that establishes the boundaries of the PI controller parameters (proportional and integral gains) that avoid any limit cycle considering that a lag-dominant first order process model is used for the tuning. By taking into account the boundaries, a safety application in an event-based framework of any tuning rule designed for the classical time-driven case is possible.

Vortex-induced vibration of bridge decks: Describing function-based model

A describing function (DF)-based model is introduced for the simulation of vortex-induced vibration (VIV) of bridge decks. Similar to the linear frequency response function, the DF is the complex ratio of the first-order component of the nonlinear output to the harmonic input. The DF can be either identified using the forced vibration technique or based on the VIV nonlinear response time history. An iterative procedure is accordingly developed to predict the VIV response with the DF-based model, and an equivalent-damping-ratio-based simplified method is further proposed to efficiently obtain the limit cycle oscillation (LCO) amplitude of VIV. It is demonstrated that the conventional van der Pol-type model is equivalent to a special case of the DF-based model for VIV. Three case studies involving various cross-sections are utilized to validate the simulation accuracy and efficiency of the proposed DF-based model for typical features at VIV lock-in such as LCO and hysteresis phenomena. The vertical VIVs can be well captured by the DF-based model, while its capability of simulating the torsional VIVs requires further improvement. Furthermore, the predictive capability of DF-based model for vertical VIVs of bridge decks within a wide range of mass-damping conditions is highlighted.

Nonlinear Stability Analysis for Three-Phase Grid-Connected PV Generators

Numerous emerging oscillatory stability issues have recently arisen as renewable energy sources (RESs) are integrated into the power system. Some of the existing literature attempts to explain these issues through the corresponding simplification, which usually omits the influence of the nonlinear discontinuous elements in the system. Hence, the obtained results are incomplete. To fill this gap, this article proposes a stability analysis method based on the describing function (DF) method to explore the influence of nonlinear discontinuous elements in the system. Without loss of generality, taking the three-phase grid-connected photovoltaic (PV) generator as the research object, the perturbation and observation (P&O)-based power control, which is a typical nonlinear discontinuous element, was carefully studied. First, the complete mathematical model of the three-phase grid-connected PV generator was established, including the outermost P&O-based power control. Then, according to the DF method, the stability of the PV generator was analyzed, and the related influence factors were studied in detail. It was found that the nonlinear discontinuous P&O-based power control has a substantial influence on the stability of the PV generator and can result in the oscillation. At the same time, the DF method and the conventional method were compared, which shows that the proposed DF method can enhance the accuracy of the stability assessment. Especially regarding critical stability, the oscillation magnitude and frequency can be calculated accurately. All the theoretical analyses were verified by the real-time hardware-in-loop (HIL) tests.

A Novel Accurate Adaptive Constant On-Time Buck Converter for a Wide-Range Operation

The ripple-based constant on-time (COT) control with ripple compensation has been widely adopted in buck converters in the recent years because of its simplicity, fast transient response, and high light-load efficiency. However, this control has the potential issues of instability and excessive load transient response with wide input/output voltage range conditions in the power management integrated circuit (PMIC) applications. In this article, an accurate adaptive COT control scheme is proposed to solve these problems. The control scheme achieves accurate output voltage, and nearly constant quality factor and well transient response over a wide-range operation with simple circuit. The small-signal model is also derived based on the describing function (DF) technique for design optimization. Finally, the simulations and experimental verifications were conducted to verify the proposed concepts.

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

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

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

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

Limit cycles in relay systems with fractional order plants

In this paper, limit cycle frequency, pulse width and stability analysis are examined using different methods for relay feedback nonlinear control systems with integer or fractional order plant transfer functions. The describing function (DF), A loci, a time domain method formulated in state space notation and Matlab/Simulink simulations are used for the analysis. Comparisons of the results of using these methods are given in several examples. In addition, the work has been extended to fractional order systems with time delay. Programs have been developed in the Matlab environment for all the theoretical methods. In particular, Matlab programs have been written to obtain a graphical solution for the A loci method, which can precisely calculate the limit cycle frequency. The developed solution methods are shown in various examples. The major contribution is to look at finding limit cycles for relay feedback systems having plants with a fractional order transfer function (FOTF). However, en route to this goal new assessments of limit cycle stability have been done for a rational plant transfer function plus a time delay.

Stability analysis of the high-order nonlinear extended state observers for a class of nonlinear control systems

The nonlinear extended state observer (ESO) is a novel observer for a class of nonlinear control system. However, the non-smooth structure of the nonlinear ESO makes it difficult to measure the stability. In this paper, the stability problem of the nonlinear ESO is considered. The describing function (DF) method is adopted to analyze the stability of high-order nonlinear ESOs. The main result of the paper shows the existence of the self-oscillation and a sufficient stability condition for high-order nonlinear ESOs. Based on the analysis results, we give a simple and fast parameter tuning method for the nonlinear ESO and the active disturbance rejection control (ADRC). Realistic application simulations show the effectiveness of the proposed parameter tuning method.

Fuzzy sliding mode control design based on stability margins

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

Analysis of Actuator Rate Limit Effects on First-Order Plus Time-Delay Systems under Fractional-Order Proportional-Integral Control

Actuator rate limit deteriorates control performance, and may even lead to system instability in precision process control. In this paper, a first-order plus time-delay (FOPTD) system class with actuator rate limit is considered. The describing function (DF) of the rate limiter is derived to obtain the describing functions of the closed-loop and open-loop systems, and the onset frequency in the Nichols chart is used to analyze the rate limit effects in frequency domain. A fractional-order proportional-integral (FOPI) controller is first designed based on the flat phase constraint, crossover frequency, and phase margin specifications. Then a traditional integer-order PID (IOPID) controller is designed based on the same specifications to compare with the FOPI controller in the presence of actuator rate saturation. A careful simulation study is presented to validate all the conclusions.

Chattering and stability analysis of the sliding mode control using inverse hyperbolic function

Chattering is a known issue in sliding mode control (SMC) and much research has been carried out to mitigate the chattering and its effects. In this work the chattering and stability analysis of the inverse hyperbolic function (IHF) based reaching law (RL) has been carried out. The work mathematically formulates the reaching time and calculates the describing function (DF) for the IHF based RL, which is used to formulate the conditions for chattering avoidance. The bound for the gain of RL has been calculated that will result in the chattering avoidance. Finally the proposed scheme has been used to control the speed of DC motor. Experimental results show avoidance of chattering and reduction in the reaching time.

Chattering Analysis of HOSM Controlled Systems: Frequency Domain Approach

In this article, a methodology of chattering analysis of Sliding Mode/Higher Order Sliding Mode (SM/HOSM) control systems in the frequency domain is presented. A numerical method for computing the Describing Functions (DFs) of HOSM control algorithms is given. The algorithm for predicted chattering parameters in dynamically perturbed system via the Describing Function-Harmonic Balance (DF-HB) technique is proposed. The stability conditions for limit cycles in dynamically perturbed HOSM control systems are presented. The concepts of Practical Stability Phase Margin ( PSPM ) and Practical Stability Gain Margin ( PSGM ) as the robustness metrics to unmodeled dynamics in HOSM control systems are introduced. The methodologies for computing PSPM and PSGM via DF-HB technique are provided. The accuracy of the proposed chattering analysis techniques is confirmed via computer simulations.

An Improved Describing Function With Applications for OTA-Based Circuits

Electronic systems make extensive use of operational transconductance amplifiers (OTAs) to build filters and oscillators. Studying the effects of the saturation nonlinearity on these OTA-based circuits is difficult and often requires lengthy simulations to check the system’s performance under large-signal operation. The describing function (DF) theory allows to circumvent these simulations by deriving a signal-dependent linearized gain, which predicts the effects of the nonlinearity. However, its use is limited since the state-of-the-art DFs deviate significantly from the real saturating behavior of OTAs. This paper proposes an improved DF, which can be directly derived from the static nonlinear characteristic of the transconductance amplifier. The performance of the proposed methodology is demonstrated for both an OTA-based filter and oscillator. It is shown that the proposed DF has a better nonlinear prediction capability than the state-of-the-art solutions.

New Characterization and Classification of Synchronization of Multiple Metronomes on a Cart via Describing Function Method

This paper aims to characterize and classify the synchronization phenomena in the system consisting of multiple identical metronomes placed on an elastically fixed cart by using describing function (DF) method, which deals with a discontinuous model of the torque driving the metronome in terms of its angle and angular velocity. Different from a previous study on the uniform amplitude synchronization of all metronomes in the system, this paper analyzes uniform and non-uniform amplitude synchronization. By investigating and using the relation between the motion of the cart (common base) and the multiple metronomes which was not studied in the previous study, this paper presents a new DF solution of the system to characterize and classify the synchronization phenomena in the system by the following two types of synchronization: One is the inherent synchronization, for which the cart is still. The frequency and amplitudes of all synchronized metronomes are the same as those of a single metronome hanging from a fixed base. The other is the non-inherent synchronization, for which the cart moves in synchronization with all metronomes. It includes in-phase uniform amplitude synchronization and non-uniform amplitude synchronization. The new characterization and classification of the synchronization is validated via numerical investigation to the system with two, three, and four metronomes.

Analysis of orbital stability in lure system, based on dynamic harmonic balance

A criterion of orbital stability of a limit cycle in a Lure system is formulated using the dynamic harmonic balance (DHB) principle and the describing function (DF) method. It is demonstrated that a more precise formulation of orbital stability than that provided by the Loeb's criterion can be produced. This enhancement is achieved through elimination of the assumption that is used in Loeb's criterion derivation. It is demonstrated in the paper that this assumption does not hold. An example of analysis is given.

Chattering existence and attenuation in fuzzy-based sliding mode control

Chattering is a low amplitude and high-frequency undesired signal which typically appears in relay or switched systems as in variable structure controllers. The chattering phenomenon can be treated as a limit cycle and the describing function (DF) method results in an useful tool for predicting its existence allowing the computation of the amplitude and frequency of such oscillations. In this paper, a sliding mode control based on fuzzy logic is proposed as an alternative to reduce chattering where DF method is used to demonstrate its attenuation level at the output of the plant. Analytical results are complemented with numerical simulations made for a pneumatic linear actuator.

Limit cycle prediction of systems with fractional controllers and backlash

This article investigates the limit cycle (LC) prediction of systems with backlash by means of the describing function (DF) when using discrete fractional-order (FO) algorithms. The DF is an approximate method that gives good estimates of LCs. The implementation of FO controllers requires the use of rational approximations, but such realizations produce distinct dynamic types of behavior. This study analyzes the accuracy in the prediction of LCs, namely their amplitude and frequency, when using several different algorithms. To illustrate this problem we use FO-PID algorithms in the control of systems with backlash.

On limit cycle chattering in sliding mode control systems under the influence of sampling intervals based on describing function approach

In this paper, the influence of sampling intervals on the chattering in sliding mode (SM) control systems is considered. The describing function (DF) approach is employed to analyze the chattering characteristics in the sampling SM control. By the DF calculations and limit cycle existence conditions, an unstable limit cycle and two stable limit cycles are found in the SM control system. The frequencies and amplitudes of the two limit cycles can also be estimated by graphical calculations. The estimation accuracy of chattering parameters is evaluated by the simulations. The results of simulations show that the system could converge to a large and a small limit cycle from different initial conditions.