Showing papers on "Describing function published in 2005"

Analysis of chattering in continuous sliding-mode controllers

Abstract: An analysis of two most popular continuous sliding-mode algorithms: The power-fractional sliding-mode algorithm and a second-order sliding-mode algorithm known as the super-twisting is carried out in the frequency domain with the use of the describing function method. It is shown that in the presence of an actuator, the transient process converges to a periodic motion. Parameters of this periodic motion are analyzed. A few examples are considered to illustrate the obtained results.

On global dynamic behavior of weakly connected oscillatory networks

Abstract: The global dynamics of weakly connected oscillatory networks is investigated: as a case study, one-dimensional arrays of third-order oscillators are considered. Through the joint application of the describing function technique and Malkin's Theorem a very accurate analytical expression of the phase deviation equation (i.e. the equation that describes the phase deviation due to the weak coupling) is derived. The total number of limit cycles and their stability properties are estimated via the analytical study of the phase deviation equation. The proposed technique significantly extends the results available in the literature and can be applied to almost all complex networks of oscillators. In particular two-dimensional, space variant and fully connected networks can be dealt with here.

Analysis of chattering in continuous sliding mode control

Abstract: An analysis of two most popular continuous sliding mode algorithms: the power-fractional sliding mode algorithm and a second order sliding mode algorithm known as the super-twisting is carried out in the frequency domain with the use of the describing function method. It is shown that in the presence of an actuator, the transient process converges to a periodic motion. Parameters of this periodic motion are analyzed. A few examples are considered to illustrate the obtained results.

Behavioral analysis of self-oscillating class D line drivers

Abstract: Self-oscillating power amplifiers (SOPAs) provide an elegant way to power signals which have high crest factors with a high efficiency. Recently, it has been shown that by pushing this concept to its limits, the stringent specifications of digital subscriber line (xDSL) could be met. For the design of these high bandwidth, low distortion amplifiers in a digital CMOS technology, a thorough analysis of the hard nonlinear system is mandatory. This paper describes behavioral models based on the describing function method and Bessel series expansion of nonlinear modulations. Models have been derived for the self-oscillation, the bandwidth, the dominant third-order distortion and all inter-modulation products of a SOPA line driver. All spectral peaks at the output of a single ended SOPA amplifier are qualitatively and quantitatively explained by these models with a very high accuracy. A calculation speed-up with three orders of magnitude could be obtained compared with a dedicated numerical simulator.

A Circuit for the Large and Small Signal Dynamic Modeling of the PRC-LCC Resonant Topology with a Capacitor as Output Filter

Abstract: This paper explains how to obtain and solve with PSpice a simple mathematical model for the series-parallel resonant topology (PRC-LCC) with a capacitor as output filter. The model provides dynamic information useful to calculate any voltage and current in the topology and design the feedback loop. If the extended describing function and the generalized averaging method are applied to the general equations of the topology, five non-linear differential equations are obtained. This five equations constitute the large signal model, which can be solved by numeric calculation or, simpler, with the aid of PSpice. To use PSpice, the large signal model of the topology is translated into an equivalent circuit. As the circuit contains all the information about dynamical behavior of the topology, large and small signal analysis can be carried out. In both cases, PSpice results have been experimentally verified. The paper can also be of use to implement the model in other simulation environments

Limit cycles in control systems employing smart actuators with hysteresis

Abstract: This paper presents theoretical and experimental results concerning a feedback control system employing a Terfenol-D-based smart actuator. Such magnetostrictive devices exhibit significant hysteresis, which, under some conditions, can generate self-sustained periodic oscillations in the control loop. The paper proposes a general procedure to find these conditions and to compute this periodic solution by exploiting some classical results about describing function analysis and the well-known Preisach operator theory. The theoretical findings are supported by a rigorous mathematical analysis resorting to fixed-point theorems.

Dynamic characterization of hysteresis elements in mechanical systems. I. Theoretical analysis.

Abstract: The pre-sliding–pre-rolling phase of friction behavior is dominated by rate-independent hysteresis. Many machine elements in common engineering use exhibit, therefore, the characteristic of “hysteresis springs,” for small displacements at least. Plain and rolling element bearings that are widely used in motion guidance of machine tools are typical examples. While the presence of a hysteresis element may mark the character of the resulting dynamics, little is to be found about this topic in the literature. The study of the nonlinear dynamics caused by such elements becomes imperative if we wish to achieve accurate control of such machines. In this Part I of the investigation, we examine a single-degree-of-freedom mass-hysteresis-spring system and show that, while the free response case is amenable to an exact solution, the more important case of forced response has no closed form solution and requires other methods of treatment. We consider harmonic-balance analysis methods (which are common analysis tools in engineering) suitable for frequency-domain treatment, in particular the approximate describing function (DF) method, and compare those results with “exact” numerical simulations. The DF method yields basically a linear equation with amplitude-dependent modal parameters. We find that agreement in the frequency response function, between DF and exact solution, is good for small excitation amplitudes and for very large amplitudes. Intermediate values, however, show high sensitivity to amplitude variations and, consequently, no regular solution is obtainable by either approach. This appears to be an inherent property of the system pointing to the need for developing further analysis methods. Experimental verification of the analysis outlined in this Part I is given in Part II of the paper.

Robust Aeroelastic Match-Point Solutions Using Describing Function Method

Abstract: A general nonlinear feedback framework to compute robust match-point solutions for aeroelastic/aeroservoelastic systems including uncertain and nonlinear operators is presented. Augmentation of the current match-point solution approach is sought using the µ method with nonlinear memoryless operators. An aeroelastic/aeroservoelastic model is built by proper interconnection of several linear fractional transformations, so that aerodynamic and structural nonlinear effects can be incorporated into a robust analysis setup. The describing function technique is used to determine the stability boundary of the nonlinear feedback system, so that limit-cycle oscillations (LCO) can be predicted and the boundary stability behavior characterized. The resulting quasilinear model allows the computation of robust stability margins that reflects both flutter and LCO instabilities. A typical section model with control surface freeplay and uncertain structural stiffness parameters is used to demonstrate the proposed aeroelastic/aeroservoelastic modeling framework. In general, good agreement was obtained with published numerical and experimental wind-tunnel test results. Comments are provided concerning possible limitations of the proposed modeling methodology.

Design of controller for buck-boost converter

Abstract: The problem of output regulation with guaranteed transient performances for buck-boost converter with inverting topology is discussed. The fast dynamical controller with the relative highest derivative of output signal in feedback loop is used. Consequently, two-time-scale motions are induced in the closed-loop system. Stability conditions imposed on the fast and slow modes and sufficiently large mode separation rate can ensure that the full-order closed-loop system achieves the desired properties in such a way that the output transient performances are desired and insensitive to external disturbances and parameter variations in the system. The existence of stable limit cycle in the fast motion subsystem gives the robustness of the output transient performances in the presence of external disturbance and parameter uncertainty. The describing function method is used to analyze the existence and parameters of stable limit cycle.

Limit cycle analysis of nonlinear sampled-data systems by gain-phase margin approach

Abstract: This work analyzes the limit cycle phenomena of nonlinear sampled-data systems by applying the methods of gain–phase margin testing, the M-locus and the parameter plane. First, a sampled-data control system with nonlinear elements is linearized by the classical method of describing functions. The stability of the equivalent linearized system is then analyzed using the stability equations and the parameter plane method, with adjustable parameters. After the gain–phase margin tester has been added to the forward open-loop system, exactly how the gain–phase margin and the characteristics of the limit cycle are related can be elicited by determining the intersections of the M-locus and the constant gain and phase boundaries. A concise method is presented to solve this problem. The minimum gain–phase margin of the nonlinear sampled-data system at which a limit cycle can occur is investigated. This work indicates that the procedure can be easily extended to analyze the limit cycles of a sampled-data system from a continuous-data system cases considered in the literature. Finally, a sampled-data system with multiple nonlinearities is illustrated to verify the validity of the procedure.

Development of vehicle dynamics model for real-time electronic control unit evaluation system using kinematic and compliance test data

Abstract: A functional suspension model is proposed as a kinematic describing function of the suspension, that represents the relative wheel displacement in polynomial form in terms of the vertical displacement of the wheel center and steering rack displacement. The relative velocity and acceleration of the wheel is represented in terms of first and second derivatives of the kinematic describing function. The system equations of motion for the full vehicle dynamic model are systematically derived by using velocity transformation method of multi-body dynamics. The comparison of test and simulation results demonstrates the validity of the proposed functional suspension modeling method. The model is computationally very efficient to achieve real-time simulation on TMS 320C6711 150 ㎒ DSP board of HILS (hardware-in-the-loop simulation) system for ECU (electronic control unit) evaluation of semi-active suspension.

A mixed time-frequency-domain approach for the analysis of a hysteretic oscillator

Abstract: The global dynamic behavior of a hysteretic oscillator is investigated. It is shown that the most significant bifurcation phenomena can be accurately detected through the joint application of two frequency-domain techniques, i.e., describing function and harmonic balance, and of a suitable time-domain method for computing limit-cycle Floquet's multipliers. The proposed approach can be effectively applied for investigating nonlinear oscillators and circuits that do not admit of a simple Lur'e representation and that exhibit a complex dynamic behavior.

Analytical Determination of Describing Function of Nonlinear Element with Fuzzy Logic

Abstract: The paper presents the procedure for analytical determination of describing function of nonlinear element with fuzzy logic nonlinearity. The procedure is based on the method of analytical determination of describing function of generalized static characteristic of nonlinear element. The procedure is also illustrated with examples.

Data-Based Robust Match-Point Solutions Using Describing Function Method

Abstract: Limit Cycle Oscillation, LCO, has been a prevalent aeroelastic/aseroservoelastic, problem on several figther aircrafts in the transonic regime. Complicated by the aircraft/stores and its flight regimes, the LCO mechanisms still remain to be fully analytically predictable. Linear flutter engineering tools only are able to predict frequency and modal composition of the LCO dynamics, but fail to predict the LCO onset as well as its oscillation amplitude. This work augments the current match-point solution approach using the μ-method analysis with nonlinear memoryless operators. These operators are characterized using a block-oriented system identification techniques. By replacing the identified memoryless operators by its related sinusoidal-input describing function, SIDF, the resulting model can be used to compute robust stability margins using the μ-method that would reflect both flutter and LCO instability regions of the flight envelope. The concept involves a representation of the aeroelastic/aeroservoelastic dynamics that includes linear, nonlinear and uncertain components using a general Linear Fractional Transformation. A general data-based modeling updating approach is sought, that is, a technique to augment existing production flutter models of actual aircraft with nonlinear operators derived by analyzing wind-tunnel/flight-test data. To illustrate this modeling framework for LCO detection, this technique is applied to a typical section with a control surface freeplay while including uncertain stiffness parameters. Comments are provided concerning possible limitations of the proposed modeling methodology.

Simple Hysteresis Describing Technique to Control a Non-Active Power Compensator

Abstract: A hysteresis control analysis technique to control an active power filter is proposed in this paper. This technique settles the maximum switching frequency of the inverter and the ripple of the compensated current as conditions for continuous cycling of the system. The describing function method is used to develop this technique that consists of the linearization of the current non-linear feedback loop. The linearization is made by deriving the hysteresis complex describing function and then applying the stability limit cycle condition to the current closed-loop to determine the frequency and amplitude values of the error signal. These values are the maximum switching frequency and the current ripple, respectively. A new "describing technique" method is proposed, which permits calculation of these parameters in a simple algebraic equation as a function of the hysteresis band, dc bus voltage and inductive low pass filter value. Moreover, the compromise between the dc bus voltage and inductor value can be evaluated easily as a function of both switching frequency and current ripple requirements. This technique is applied to design a single-phase non-active power compensator. Simulation results have shown that the predicted values are very close to the values obtained by simulation

Weakly connected oscillatory networks for dynamic pattern recognition

Abstract: Many studies in neuroscience have shown that nonlinear dynamic networks represent a bio-inspired models for information and image processing. Recent studies on the thalamo-cortical system have shown that weakly connected oscillatory networks have the capability of modelling the architecture of a neurocomputer. In particular they have associative properties and can be exploited for dynamic pattern recognition. In this manuscript the global dynamic behavior of weakly connected cellular networks of oscillators are investigated. It is assumed that each cell admits of a Lur'e description. In case of weak coupling the main dynamic features of the network are revealed by the phase deviation equation (i.e. the equation that describes the phase deviation due to the weak coupling). Firstly a very accurate analytic expression of the phase deviation equation is derived via the joint application of the describing function technique and of Malkin's Theorem. Then it is shown that the total number of periodic limit cycles with their stability properties can be estimated through the analysis of the phase deviation equation.

On global dynamic behavior of weakly connected cellular nonlinear networks

Abstract: It is shown that the global dynamics of weakly connected cellular nonlinear networks can be investigated through the joint application of Malkin's theorem and of the describing function technique. As a case study a one-dimensional array of third order oscillators is considered. Firstly a very accurate analytical expression of the phase deviation equation (i.e. the equation that describes the phase deviation due to the weak coupling) is derived. Then the total number of limit cycles and their stability properties are estimated via the analytical study of the phase deviation equation. We remark that the proposed technique can be applied to a large class of weakly connected nonlinear networks. In particular two-dimensional, space variant and fully connected networks can be dealt with.

A State-Space Modeling Approach for Dynamic Nonlinear Microwave System Based on Large Signal Time Domain Measurements

Abstract: This paper introduces a model for dynamic nonlinear microwave systems or device operating in large signal regime, based on a state-space formulation. It differs from previous approaches in the comprehensive description of the describing functions and the identification procedure based on the solution of a linear system of equations. The model is able to predict specific operation states, having identified the model in an a-priory defined set of operation states. The identification procedure, as well as the validation, are based on multisine Large Signal Time Domain Measurements at different input power levels. Extensive experimental results are provided for a case study based on a commercial InGaP HBT gain block driven by a set of 9 tones equally distributed in 1.6 MHz bandwidth around the frequency of 4.9GHz, with all the power levels ranging from linear to saturation regions. The accuracy of the model, also in terms of prediction capabilities is provided in both time- and frequency-domain.

Simultaneous Closed-loop Identification of Nonlinear and Linear part of a Hammerstein-type Nonlinear Model

Abstract: A procedure for the closed-loop identification of some class of non-linear plants of Hammerstein type, based on a generalization of the well-known Ziegler-Nichols' (ZN) experiment, is proposed It may be used for the simultaneous estimation of the plant dynamic and the describing function of the plant nonlinearity As in the ZN method, no extra equipment is needed for performing the experiment

Stability of Haptic Rendering: Discretization,

Abstract: Rendering stiff virtual objects remains a core chal- lenge in the field of haptics. A study of this problem is presented, which relates the maximum achievable object stiffness to the elements of the control loop. In particular, we examine how the sampling rate, quantization, computational delay, and amplifier dynamics interact with the inertia, natural viscous, and Coulomb damping of the haptic device. Nonlinear effects create distinct stability regions and many common devices operate stably, yet in violation of passivity criteria. An energy based approach provides theoretical insights, supported by simulations, experimental data, and a describing function analysis. The presented results subsume previously known stability conditions.

Real-time Parameter Estimation of Hydraulic Servo Actuator Systems Using Self-excited Oscillation Method : Online Identification of Approximated Transfer Function Composed of Integral and Second Order Lag Elements

Abstract: The hydraulic servo actuator system is generally dealt with as a second order delay element. The self-excited oscillation method has the advantage that it enables easy estimation of the dynamic parameters of the approximated second order transfer function without valuable measuring instruments such as a FFT analyzer. This study utilizing the self-excited oscillation method suggests the real-time identification algorithm. In order to demonstrate the method's effectiveness, the proposed method was experimentally compared with the frequency response characteristics. Results indicate that both method shows good coincident. It was also confirmed that when the supply pressure and additional torque are continuously changed in the hydraulic system, the damping coefficient and undamped natural frequency were updated on the PC monitor. In addition, amplitude and frequency correction coefficients are analytically obtained from the describing function considering the phase shift, and compared with the simulation results.

Practical frequency response analysis of non-linear time-delayed differential or difference equation models

Abstract: A systematic algorithm for generating the polyharmonic balance equations for any system within a broad class of time-delayed differential, or non-linear difference equation models, is presented. The method, which is readily automated, enables the balance equations to be written down directly in terms of the coefficients of the governing equation, and the complex amplitudes of a general harmonic waveform. The system frequency response or amplitude dependent describing function is then readily computed. The method is illustrated by means of examples including both a time-delayed differential system example and a discrete time NARX model application. The results are validated against detailed numeric simulation which confirms the accuracy and efficacy of the approach.

An identification technique for macromodeling of long interconnects

Abstract: We consider the problem of the identification of describing functions for the reduced time domain modeling of interconnects. In particular we identify the rational approximation of regular parts, once delays are exactly extracted, with a combination of vector fitting algorithm with a post-processing procedure based on a non linear least square refinement step. This improves the accuracy of the approximation obtained, and leads directly to rank-1 residue matrices. The technique is applied to a couple of test cases and a comparative evaluation of performances is carried out.

Nonlinear loop shaping for suppressing sensitivity hump in hard disk drive servo systems

Abstract: In hard disk drive (HDD) servo system that uses voice coil motor (VCM) as the actuator, the sensitivity hump is unavoidable according to Bode's integral theorem (BIT). That results in amplification of disturbances with frequencies beyond bandwidth. This paper proposes a nonlinear loop shaping method to suppress sensitivity hump of the conventional servo. The proposed nonlinear algorithm consists of two parts. One is the conventional track-following control law, the other is a nonlinear PD type control law designed based on the conventional one to shape its high frequency response. The simulated describing function shows that sensitivity peak can be suppressed with the proposed algorithm without degrading low frequency disturbance rejection, which could lead to increase the track-following accuracy. The efficacy is verified through vibration test with disk flutter. Although sensitivity is improved, the effect of high frequency measurement noise on the position error signal (PES) which is governed by complementary sensitivity function is not degraded. The robustness is also verified with high frequency plant uncertainties.

Algebraic representation of error bounds for describing function using Groebner base [nonlinear circuit analysis example]

Abstract: This paper presents an algebraic approach to find the region where the true periodic solution of a nonlinear system or circuit lies when a describing function solution is given. Because algebraic representations of the error bounds are obtained by the Groebner base, the dependence of the bounds on parameter values becomes clear. Further, we propose an efficient method to improve the estimation. The Groebner base is shown to be applicable to the describing function method. The estimation is modified considerably.

Parameter plane analysis of fuzzy vehicle steering control systems

Abstract: The main purpose of this paper is to analyze the robust stability for a fuzzy vehicle steering control system. In general, fuzzy control system is a nonlinear control system. Therefore, the fuzzy controller may be linearized by the use of describing function first. After then, parameter plane method is then applied to determine the conditions of robust stability when the system has perturbed or adjustable parameters. A systematic procedure is proposed to solve this problem. The effects of plant parameters and control factors are both considered here. Furthermore, the problem of relative stability by using gain-phase margin tester is also addressed. The limit cycles provided by a static fuzzy controller can be easily suppressed if the control factors are chosen properly. Simulation results show the efficiency of our approach.