Showing papers on "Describing function published in 2015"

Modeling and Control of Series–Series Compensated Inductive Power Transfer System

Abstract: In this paper, a T-equivalent circuit for loosely coupled transformer, which does not involve magnetic coupling, is presented. A detailed dynamic analysis based on extended describing function technique is presented for a series–series compensated inductive power transfer system. The continuous-time large-signal model, the steady-state operating point, and the small-signal model are derived in an analytical closed-form. This model includes both the frequency and the phase-shift control. Simulation and experimental verification results of the derived models are presented to validate the presented analysis.

Design and Modeling of Anti Wind Up PID Controllers

Abstract: In this chapter several anti windup control strategies for SISO and MIMO systems are proposed to diminish or eliminate the unwanted effects produced by this phenomena, when it occurs in PI or PID controllers. Windup is a phenomena found in PI and PID controllers due to the increase in the integral action when the input of the system is saturated according to the actuator limits. As it is known, the actuators have physical limits, for this reason, the input of the controller must be saturated in order to avoid damages. When a PI or PID controller saturates, the integral part of the controller increases its magnitude producing performance deterioration or even instability. In this chapter several anti windup controllers are proposed to eliminate the effects yielded by this phenomena. The first part of the chapter is devoted to explain classical anti windup architectures implemented in SISO and MIMO systems. Then in the second part of the chapter, the development of an anti windup controller for SISO systems is shown based on the approximation of the saturation model. The derivation of PID SISO (single input single output) anti windup controllers for continuous and discrete time systems is implemented adding an anti windup compensator in the feedback loop, so the unwanted effects are eliminated and the system performance is improved. Some illustrative examples are shown to test and compare the performance of the proposed techniques. In the third part of this chapter, the derivation of a suitable anti windup PID control architecture is shown for MIMO (multiple input multiple output) continuous and discrete time systems. These strategies consist in finding the controller parameters by static output feedback (SOF) solving the necessary linear matrix inequalities (LMI’s) by an appropriate anti windup control scheme. In order to obtain the control gains and parameters, the saturation is modeled with describing functions for the continuous time case and a suitable model to deal with this nonlinearity in the discrete time case. Finally a discussion and conclusions sections are shown in this chapter to analyze the advantages and other characteristics of the proposed control algorithms explained in this work.

Observer-Based Control of LLC DC/DC Resonant Converter Using Extended Describing Functions

Abstract: This paper presents theoretical and practical results about dynamic analysis, frequency response, and control of a LLC resonant dc/dc converter operating under wide input voltage and load variations. A nonlinear model for the LLC resonant converter was developed using the extended describing function method; then, based on the derived model, a nonlinear observer-based controller was designed and implemented with a digital signal processor. Transient responses obtained under input voltage and output load variations show that the proposed controller is capable to stabilize the output effectively. Experimental results prove the superiority of the proposed observer-based controller over a conventional PID controller.

Nonlinear damping and quasi-linear modelling.

Abstract: The mechanism of energy dissipation in mechanical systems is often nonlinear. Even though there may be other forms of nonlinearity in the dynamics, nonlinear damping is the dominant source of nonlinearity in a number of practical systems. The analysis of such systems is simplified by the fact that they show no jump or bifurcation behaviour, and indeed can often be well represented by an equivalent linear system, whose damping parameters depend on the form and amplitude of the excitation, in a 'quasi-linear' model. The diverse sources of nonlinear damping are first reviewed in this paper, before some example systems are analysed, initially for sinusoidal and then for random excitation. For simplicity, it is assumed that the system is stable and that the nonlinear damping force depends on the nth power of the velocity. For sinusoidal excitation, it is shown that the response is often also almost sinusoidal, and methods for calculating the amplitude are described based on the harmonic balance method, which is closely related to the describing function method used in control engineering. For random excitation, several methods of analysis are shown to be equivalent. In general, iterative methods need to be used to calculate the equivalent linear damper, since its value depends on the system's response, which itself depends on the value of the equivalent linear damper. The power dissipation of the equivalent linear damper, for both sinusoidal and random cases, matches that dissipated by the nonlinear damper, providing both a firm theoretical basis for this modelling approach and clear physical insight. Finally, practical examples of nonlinear damping are discussed: in microspeakers, vibration isolation, energy harvesting and the mechanical response of the cochlea.

Discrete‐time network‐based control under scheduling and actuator constraints

Abstract: In Chaps. 2– 8, we have studied networked control for continuous-time plants. Chapters 9– 12 will present our analysis on networked control for discrete-time plants. In this chapter, we consider discrete-time NCSs with communication constraints and actuator constraints. Systems with actuator constraints were extensively studied during the 1960s due to their intimate connection with optimal control. Concurrently, the design approaches, such as the describing function method, which dealt specifically with nonlinearities such as saturation were developed. Only very limited research into actuator saturation was carried out during the 1970s and 1980s with the emphasis being placed mostly on the development of the linear state space approach and its numerous offshoots.

Absolute stability analysis of non-linear active disturbance rejection control for single-input–single-output systems via the circle criterion method

Abstract: This study focuses on the stability analysis of non-linear active disturbance rejection control (ADRC) for single-input–single-output systems. Firstly, a non-linear ADRC system for a linear plant is transformed into a Lurie system. Secondly, two extended circle criteria are obtained, and two numerical examples are presented to illustrate the absolute stability analysis, including both stable and unstable linear plants. Thirdly, local asymptotic stability of a non-linear ADRC system for a non-linear plant is also performed through linearisation by Taylor expansion. Finally, a comparison with the existed processing methods is further made, including the describing function method and time domain stability analysis method. It can be concluded that the circle criterion method is more convenient and practical for its frequency domain and graphical interpretation. The circle criterion method can also be extended to the stability analysis of a control system which applies linear ADRC to a plant with one non-linear term.

Frequency domain stability analysis of nonlinear active disturbance rejection control system

Abstract: This paper applies three methods (i.e., root locus analysis, describing function method and extended circle criterion) to approach the frequency domain stability analysis of the fast tool servo system using nonlinear active disturbance rejection control (ADRC) algorithm. Root locus qualitative analysis shows that limit cycle is generated because the gain of the nonlinear function used in ADRC varies with its input. The parameters in the nonlinear function are adjustable to suppress limit cycle. In the process of root locus analysis, the nonlinear function is transformed based on the concept of equivalent gain. Then, frequency domain description of the nonlinear function via describing function is presented and limit cycle quantitative analysis including estimating prediction error is presented, which virtually and theoretically demonstrates that the describing function method cannot guarantee enough precision in this case. Furthermore, absolute stability analysis based on extended circle criterion is investigated as a complement.

Advanced controller design for a series-series compensated inductive power transfer charging infrastructure using asymmetrical clamped mode control

Abstract: Wireless charging of electric vehicles require a significant air gap between the primary and secondary winding of an inductive power transfer (IPT) system. Due to the existence of the air gap, power flow regulation to keep the output voltage constant becomes a non-trivial task. Hence, the bandwidth, phase margin, and gain margin of the voltage control loops should be appropriately designed, in order to guarantee a robust system. In this paper, a generalized small-signal modelling of series-series compensated (SS topology) IPT system using extended describing function concept has been presented. Using this small-signal model, a controller has been designed for fixed frequency and variable duty cycle, to control the output voltage. Since an asymmetrical clamped mode control (ACM) requires a lower switching frequency compared to the popular fixed frequency control strategies, viz. symmetrical clamed mode control (SCM) and asymmetrical duty cycle control (ADC), it has been used to control the output voltage.

Small-signal equivalent circuit model of series resonant converter

Abstract: A simple third-order equivalent circuit model of series resonant converter (SRC) is proposed in this paper. Up to now, the most successful equivalent circuit model of SRC is based on extended describing function concept, which is proposed by Dr. E. Yang [30]. However, the equivalent circuit is a complicated fifth-order circuit with the cross-coupling effect and no analytical solution is provided for transfer functions. This paper proposes a methodology to simplify the fifth-order equivalent circuit to a third-order equivalent circuit. The equivalent circuit model can predict the dynamic behavior very well when switching frequency is below, close to or above resonant frequency. Furthermore, for the first time, analytical expressions of transfer functions are provided to serve as a useful tool for feedback design. The equivalent circuit model is verified by Simplis simulation and experimental results.

Stochastic linearisation approach to performance analysis of feedback systems with asymmetric nonlinear actuators and sensors

Abstract: This paper considers feedback systems with asymmetric (i.e., non-odd functions) nonlinear actuators and sensors. While the stability of such systems can be investigated using the theory of absolute stability and its extensions, the current paper provides a method for their performance analysis, i.e., reference tracking and disturbance rejection. Similar to the case of symmetric nonlinearities considered in earlier work, the development is based on the method of stochastic linearisation (which is akin to the describing functions, but intended to study general properties of dynamics, rather than periodic regimes). Unlike the symmetric case, however, the nonlinearities considered here must be approximated not only by a quasilinear gain, but a quasilinear bias as well. This paper derives transcendental equations for the quasilinear gain and bias, provides necessary and sufficient conditions for existence of their solutions, and, using simulations, investigates the accuracy of these solutions as a tool for pre...

Adaptive control of Hammerstein systems with unknown Prandtl–Ishlinskii hysteresis

Abstract: We numerically investigate the sense in which an adaptive control law achieves internal model control of Hammerstein plants with Prandtl-Ishlinskii hysteresis. We apply retrospective cost adaptive control to a command-following problem for uncertain Hammerstein systems with hysteretic input nonlinearities. The only required modeling information of the linear plant is a single Markov parameter. Describing functions are used to determine whether the adaptive controller inverts the plant at the exogenous frequencies.

Reduced fractal model for quantitative analysis of averaged micromotions in mesoscale: Characterization of blow-like signals

Abstract: It has been shown that many micromotions in the mesoscale region are averaged in accordance with their self-similar (geometrical/dynamical) structure. This distinctive feature helps to reduce a wide set of different micromotions describing relaxation/exchange processes to an averaged collective motion, expressed mathematically in a rather general form. This reduction opens new perspectives in description of different blow-like signals (BLS) in many complex systems. The main characteristic of these signals is a finite duration also when the generalized reduced function is used for their quantitative fitting. As an example, we describe quantitatively available signals that are generated by bronchial asthmatic people, songs by queen bees, and car engine valves operating in the idling regime. We develop a special treatment procedure based on the eigen-coordinates (ECs) method that allows to justify the generalized reduced fractal model (RFM) for description of BLS that can propagate in different complex systems. The obtained describing function is based on the self-similar properties of the different considered micromotions. This kind of cooperative model is proposed here for the first time. In spite of the fact that the nature of the dynamic processes that take place in fractal structure on a mesoscale level is not well understood, the parameters of the RFM fitting function can be used for construction of calibration curves, affected by various external/random factors. Then, the calculated set of the fitting parameters of these calibration curves can characterize BLS of different complex systems affected by those factors. Though the method to construct and analyze the calibration curves goes beyond the scope of this paper, this result could benefit future studies that will employ the developed reduced models in diagnosis, prevention, and control of unpredicted and undesired phenomena of some engineering applications that possibly exhibit such BLS.

Chaos in PID Controlled Nonlinear Systems

Abstract: Controlling nonlinear systems with linear feedback control methods can lead to chaotic behaviors. Order increase in system dynamics due to integral control and control parameter variations in PID controlled nonlinear systems are studied for possible chaos regions in the closed-loop system dynamics. The Lur’e form of the feedback systems are analyzed with Routh’s stability criterion and describing function analysis for chaos prediction. Several novel chaotic systems are generated from second-order nonlinear systems including the simplest continuous-time chaotic system. Analytical and numerical results are provided to verify the existence of the chaotic dynamics.

A novel stability analysis approach based on describing function method using for DC-DC converters

Abstract: Due to the nonlinearity of the switching process in DC-DC converters, the stability analysis with traditional linear modelling for DC-DC converters are invalid sometimes, especially in the transition interval from stable region to unstable region. In this paper, describing function method is firstly adopted in switching process modelling of DC-DC converters. Further, a new model based on describing function method for switching process is proposed to investigate the stability of the DC-DC converters. A boost converter is taken as an example in this paper, its traditional linear model and the proposed model with describing function method are both established, and the stability analysis based on these two kinds of models are carried out, respectively. Different stability analysis results are obtained according to the two kinds of models for the same boost converter with same circuit parameters and control parameters. According to the simulation and experimental results, it can be concluded that the stability analysis based on the proposed model with describing function method is more accurate and effective in the transition interval from stable region to unstable region, which provide a new stability analysis way for DC-DC converters.

Design of a variable gain integrator with reset

Abstract: This paper studies the properties of a variable gain integrator with reset, i.e. a nonlinear lag filter that is obtained by a) saturating the input, b) filtering the saturated input with a Clegg integrator, and c) add the filtered output to the unsaturated input before applying it to a PID-based controller. Depending on the amount of saturation, the corner frequency of the lag filter is reduced along with the associated phase lag. This follows from a describing function analysis in which at low frequencies a minus 20 dB/decade amplitude decay is realized with a phase lag of only 32.48 degrees. Conditions to assess global asymptotic stability of the closed-loop nonlinear control system are provided that are based on a circle criterion-like argument for the flow condition, which applies to the intervals without resets, combined with a jump condition at reset. The reset integrator design is demonstrated on a piezo-actuated motion system where its favorable phase and amplitude properties induce overshoot and settling times comparable to a single (linear) integrator, but with the disturbance rejection properties of a double integrator.

Self-Oscillations in Dynamic Systems: A New Methodology via Two-Relay Controllers

Abstract: Introduction.- Part I: Design of Self-Oscillations using Two-Relay Controller.- Describing Function-Based Design of TRC for Generation of Self-Oscillation.- Poincare Maps Based Design.- Self-Oscillation via Locus of a Perturbed Relay System Design (LPRS).- Part II: Robustification of the Self-Oscillation Generated by Two-Relay Controller.- Robustification of the Self-Oscillation via Sliding Modes Tracking Controllers.- Output-Based Robust Generation of Self-Oscillations.- Part III: Applications.- Generating Self-Oscillations in Furuta Pendulum.- Three Link Serial Structure Underactuated Robot.- Generation of Self-Oscillations in Systems with Double Integrator.- Fixed-Phase Loop (FPL).- Appendix A: Describing Function.- Appendix B: The Locus of a Perturbed Relay System (LPRS).- Appendix C: Poincare Map.- Appendix D: Output Feedback.- References.- Index.

State-space realization of a describing function

Abstract: The describing function is a powerful tool for characterizing nonlinear dynamical systems in the frequency domain. In some cases, it is the only available description of a nonlinear operator characterizing a certain subcomponent of the system. This paper presents a methodology to provide a state-space realization of one given describing function, in order to allow the study of the system in the time domain as well. The realization is based on Hammerstein models and Fourier–Bessel series. It can be embedded in time domain simulations of complex configurations with many nonlinear elements interacting, accurately describing the nonlinear saturation of the system. The technique is applied to an example application in the field of combustion instability, featuring self-excited thermoacoustic oscillations. We benchmark the performance of the tool comparing the results with a frequency domain analysis of the same system, obtaining good agreement between the two formulations.

Influence of Jitter on Limit Cycles in Bang-Bang Clock and Data Recovery Circuits

Abstract: In bang-bang (BB) clock and data recovery circuits (CDR) limit cycles can occur, but these limit cycles are undesired for a good operation of the BB-CDR. Surprisingly, however, a little bit of noise in the system is beneficial, because it will quench the limit cycles. Until now, authors have always assumed that there is enough noise in a BB-CDR such that no limit cycle occurs. In this work, a pseudo-linear analysis based on describing functions is used to investigate this. In particular, the relationship between the input noise and the amplitude of eventual limit cycles is investigated. An important result of the theory is that it allows to quantify the influence of the different loop parameters on the minimal amount of input jitter needed to destroy the limit cycle. Additionally, for the case that there is not enough noise, the worst case amplitude of the limit cycle (which is unavoidable in this case) is quantified as well. The presented analysis exhibits excellent matching with time domain simulations and leads to very simple analytical expressions.

Disturbance attenuation for systems with second-order sliding modes via linear compensators

Abstract: Non-ideal disturbance attenuation emerges in systems with sliding mode controllers when parasitic dynamics are present. In this work, the problem of non-ideal disturbance attenuation in systems controlled by second-order sliding modes is considered. Describing function and locus of a perturbed relay system (LPRS) methods are used for analysis of non-ideal disturbance attenuation. The proposed solution is to introduce a linear compensator (series or parallel) in the control loop to modify the system frequency response, that is, the Nyquist plot or LPRS. The design methodology for both parallel and series compensators is presented. The proposed solution allows one to improve disturbance attenuation and tracking quality, thereby reducing the effect of parasitic dynamics which is reflected in the system frequency response. Experiments supporting the proposed approach and simulations are provided.

Prediction of limit cycles in nonlinear systems with reset controllers using describing function

Abstract: In nonlinear systems, limit cycles may exsist in some's parameters and may cause some unexpected damage. This paper proposes a design method of a reset controller for the Reset control of limit cycles. We consider a plant consisting of a linear system with a nonlinear feedback element and show a condition of the parameter for which limit cycles exists, using describing functions of the nonlinear element and the reset controller. The adjustable parameter of the reset controller is determined by the intersection of the inverse Nyquist locus of the linear system and the existence region of the limit cycles on the complex plain. We apply the proposed design method of the reset controller to the van der Pol equation and show its effectiveness.

A Preliminary Study on Piezo-aeroelastic Energy Harvesting Using a Nonlinear Trailing-Edge Flap

Abstract: Recently, piezo-aeroelastic energy harvesting has received greater attention. In the present study, a piezo-aeroelastic energy harvester using a nonlinear trailing-edge flap is proposed, and its nonlinear aeroelastic behaviors are investigated. The energy harvester is modeled using a piezo-aeroelastic model of a two-dimensional typical section airfoil with a trailing-edge flap (TEF). A piezo-aeroelastic analysis is carried out using RL and time-integration methods, and the results are verified with the experimental data. The linearizing method using a describing function is used for the frequency domain analysis of the nonlinear piezo-aeroelastic system. From the linear and nonlinear piezo-aeroelastic analysis, the limit cycle oscillation (LCO) characteristics of the proposed energy harvester with the nonlinear TEF are investigated in both the frequency and time domains. Finally, the authors discuss the air speed range for effective piezo-aeroelastic energy harvesting.

On multiple limit cycles in sliding-mode control systems via a generalized describing function approach

Abstract: This paper proposes a generalized describing function (DF) approach to analyze a class of frequency-dependent nonlinearities which are hard to deal with by traditional DF approaches. The proposed approach can intuitively predict the number, stability, and parameters of limit cycles by graphical illustrations of amplitude–phase characteristics. When applied to a 2-relative- degree sampling output feedback system, multiple limit cycles are found in the sliding-mode control. Both behaviors of complex limit cycles and influences of initial conditions on limit cycle oscillations are analyzed. Results show that each limit cycle has its own attraction region which can be estimated by the chattering amplitudes of the state variables, and initial conditions in different attraction regions may induce multiple limit cycles in the same control system. The undesired limit cycles can be eliminated by parameter tuning, such as control gains and sampling periods, to obtain a globally stable one. Simulations and hardware experiments further confirm the validity of the generalized DF approach.

A New Perspective on the Flame Describing Function of a Matrix Flame

Abstract: This paper considers a fundamental thermoacoustic test rig developed by Noiray (“Linear and nonlinear analysis of combustion instabilities, application to multipoint injection systems and control strategies”, PhD thesis, Ecole Centrale Paris, 2007) and models it with an entirely analytical approach. The test rig is treated as a system of two coupled elements: an acoustic resonator and a flame with oscillating rate of heat release. We describe the acoustics of the combustion rig in terms of modes, and derive a governing equation for one such mode. This turns out to be the equation for a damped harmonic oscillator, forced by the heat release rate from the flame. In order to model the heat release rate, and in particular its nonlinear aspects, we develop a generalised nτ-law with amplitude-dependent coefficients and multiple time-lag. The coefficients are determined from Noiray’s measured flame describing function. Stability predictions are made by evaluating the sign of the damping coefficient in the govern...

Analysis of Nonlinear Missile Guidance Systems Through Linear Adjoint Method

Abstract: In this paper, a linear simulation algorithm, the adjoint method, is modified and employed as an efficient tool for analyzing the contributions of system parameters to the miss - distance of a nonlinear time-varying missile guidance system model. As an example for the application of the linear adjoint method, the effect of missile flight time on the miss - distance is studied. Since the missile model is highly nonlinear and a time-varying linearized model is required to apply the adjoint method, a new technique that utilizes the time-reversed linearized coefficients of the missile as a replacement for the time-varying describing functions is applied and proven to be successful. It is found that, when compared with Monte Carlo generated results, simulation results of this linear adjoint technique provide acceptable accuracy and can be produced with much less effort.

Novel controller design for plants with relay nonlinearity to reduce amplitude of sustained oscillations: Illustration with a fractional controller.

Abstract: This paper proposes a novel constrained optimization problem to design a controller for plants containing relay nonlinearity to reduce the amplitude of sustained oscillations. The controller is additionally constrained to satisfy desirable loop specifications. The proposed formulation is validated by designing a fractional PI controller for a plant with relay.

Fractional order controller for satellite attitude control system with PWPF modulator

Abstract: A nonlinear flexible satellite attitude control system with a single degree-of-freedom is considered in the framework of fractional order calculus. A pulse-width pulse-frequency (PWPF) modulated thruster is the main actuator in the system. To avoid interaction of the pulse modulation with the flexible structure modes, a new method — fractional describing function method is proposed to describe the nonlinear dynamics of PWPF actuators. Thus, the plant is considered as a fractional order system and a fractional order [PD] controller is synthesized for the rigid body. The fractional order controller is robust to resonances with high frequencies. Both bandwidth and phase margin of the system with proposed fractional order controller are larger than that of system with traditional lead-phase compensator. The fractional order modeling and design method developed here is practical and simple.