Showing papers on "Describing function published in 2004"

Modeling of quantization effects in digitally controlled DC-DC converters

Abstract: In digitally controlled DC-DC converters with a single voltage feedback loop, the two quantizers, namely the A/D converter and the digital pulse-width modulator (DPWM), can cause undesirable limit-cycle oscillations. In this paper, static and dynamic models that include the quantization effects are derived and used to explain the origins of limit-cycle oscillations. In the static model, existence of DC solution, which is a necessary no-limit-cycle condition, is examined using a graphical method. A concept of amplitude and offset dependent gain is introduced to extend the describing function method and derive the dynamic system model. From the static and dynamic models, no-limit-cycle conditions associated with A/D, DPWM and compensator design criteria are derived. The conclusions are illustrated by simulation and experimental examples.

Analysis of second-order sliding-mode algorithms in the frequency domain

Abstract: A frequency domain analysis of the second-order sliding-mode algorithms, particularly of the twisting algorithm is carried out in the frequency domain with the use of the describing function method and Tsypkin's approach. It is shown that in the presence of an actuator, the transient process may converge to a periodic motion. Parameters of this periodic motion are analyzed. A comparison of the periodic solutions in the systems with higher order sliding-mode controllers and the oscillations that occur in classical sliding-mode systems with actuators is done.

Theoretical analysis of the dynamic behavior of hysteresis elements in mechanical systems

Abstract: Many machine elements in common engineering use exhibit the characteristic of “hysteresis springs”. Plain and rolling element bearings that are widely used in motion guidance of machine tools are typical examples. The study of the non-linear dynamics caused by such elements becomes imperative if we wish to achieve accurate control of such machines. This paper outlines the properties of rate-independent hysteresis and shows that the calculation of the free response of a single-degree-of-freedom (SDOF) mass-hysteresis-spring system is amenable to an exact solution. The more important issue of forced response is not so, requiring other methods of treatment. We consider the approximate describing function method and compare its results with exact numerical simulations. Agreement is good for small excitation amplitudes, where the system approximates to a linear mass-spring-damper system, and for very large amplitudes, where some sort of mass-line is approached. Intermediate values however, show high sensitivity to amplitude variations, and no regular solution is obtained by either approach. This appears thus to be an inherent property of the system pointing to the need for developing further analysis methods.

Periodic oscillations and bifurcations in cellular nonlinear networks

Abstract: A spectral technique is proposed for investigating periodic oscillations and bifurcations in cellular nonlinear networks. The method consists of the following three fundamental steps: 1) an accurate estimation of the whole set of stable and unstable limit cycles is provided through the application of the describing function technique; 2) a detailed characterization of each limit cycle is obtained via the harmonic balance (HB) technique, by exploiting as input parameter the single harmonic approximation, provided by the describing function technique; 3) limit cycle stability and bifurcations are studied by computing the Floquet's multipliers and exploiting both a time-domain and an HB-based technique.

Non-linear modal analysis methods for engineering structures

Abstract: This thesis presents two novel nonlinear modal analysis methods, aimed at the identification of representative engineering structures. The overall objective is to detect, localize, identify and quantify the nonlinearities in large systems, based on nonlinear frequency response functions (FRFs) as input data. The methods are first introduced in a direct-path, by analyzing a general theoretical system. Then, the concepts are extended to tackle a nonlinear identification via the reverse-path of the same methodologies. The nonlinear formulation of this work is based in first-order describing functions, which represent the nonlinearities by amplitudedependent coefficients. This formulation is the basic “engine” of the methods and techniques developed here. For the sake of clarity, the research has been restricted to deal with cubic stiffness and friction damping nonlinearities, although the inclusion of other types should be straightforward, given the generality of the developments. The first direct-path method, the so-called “explicit formulation” (EF), is conducted entirely in the physical domain. This technique manipulates the physical coefficients stored in the system matrices, thus the term “explicit”, yielding the nonlinear FRF at a selected DOF as a closed-form expression, regardless of the system’s size. An optimized version of this method has been validated against real measurements taken from a test rig, and it was found that the nonlinear behaviour was predicted with reasonable accuracy. A reverse path of the “explicit formulation”, REF, was implemented as a nonlinear identification tool. In spite of successful results, it was concluded that the computational cost of this approach was too high to gain acceptance in a practical analysis. Still, the method provides a much needed bridge between a full-size theoretical model and the relatively small number of experimental measurements that may be available. The second main method, operating in a direct-path, is called the “hybrid modal technique” (HMT). It is based on a novel nonlinear modal expansion, which is analogous to existing nonlinear modal superposition techniques. The underlying linear system is expressed in generalized modal coordinates, while the nonlinearities are kept in the physical domain. The use of hybrid coordinates is a central feature, by which the localization of the nonlinearities is fully addressed. A reverse-path of this method, R-HMT, incorporates the successive application of several “standalone” techniques, also developed here, which can be used independently to tackle different aspects of nonlinear modal analysis. When gathered together, the individual techniques provide a robust methodology, able to perform a nonlinear identification within the usual experimental restrictions, while exhibiting high computational efficiency. The type of the nonlinearity can be identified by a newly introduced technique, based in the geometrical “footprint” of the extracted nonlinear component. The localization of the nonlinearities is then achieved by a linear least-squares calculation over a predefined nonlinear region of arbitrary size. This technique provides an unambiguous localization, provided that the analyzed frequency range is a fair representation of the system. Although the nonlinear natural frequencies and modal damping are not explicitly needed for identifying the system or regenerating the responses at some other forcing level, a fast approximation technique (FAT) is introduced, allowing the analytical derivation of these parameters via newly-developed expressions. The FAT establishes links with other nonlinear methods and standard linear modal analysis techniques. I would like to dedicate this thesis to: My daughter, who is on her way right now... Rosca, for loving and supporting me no matter what... My parents, J. Hugo & Delia, for absolutely everything else... Ana Sofia arrived on July 31st, 2004.

Nonlinear Aeroelastic Analysis of a Deployable Missile Control Fin

Abstract: The nonlinear aeroelastic characteristics of a deployable missile control fin have been investigated. Modes from free vibration analysis and a doublet-point method are used for the computation of supersonic unsteady aerodynamic forces. The minimum-state approximation is used to approximate the aerodynamic forces. The fictitious mass modal approach is applied to reduce the problem size and the computation time in the linear and nonlinear flutter analyses. For the nonlinear flutter analysis, the deployable hinge is represented by an asymmetric bilinear spring and is linearized using the dual-input describing function method. From the nonlinear flutter analysis, three different types of limit-cycle oscillations are observed in the wide range of airspeed over the linear flutter boundary. The aeroelastic characteristics of the missile control fin can become more stable due to the existence of the deployable hinge nonlinearity.

Applications of Robust Control to Nonlinear Systems

Abstract: The Describing Function General Describing Function Evaluation Methods The Inverse Describing Function Optimal Control Performance Specification Control Synthesis Riccati Solution for an Augmented Plant Containing a Describing Function Robustness Analysis via Simplicial Algorithms Analytic Geometry Simplicial Mapping Simplex Nulling Integer Labeling Vector Labeling Nonlinear Control Approaches to Nonlinear Control Control of a System with a Relay Element via Loop-Shifting Control via an Adaptive Perturbation Filter Nonlinear Robustness Analysis of a Relay Element via Simplicial Algorithms Direct Solution using the Two-Riccati Equation Approach Nonlinear Control of a Missile's Roll Axis Computer Algorithms Optimization Fortran Simulation Variable Dimension Restart Algorithm Hardware Implementations

Design and analysis of SISO fuzzy logic controller for power electronic converters

Abstract: Power converters are non-linear systems that usually employ linear controllers designed to offer good small signal performance at the nominal operating point. Yet, the converter's large-signal response is generally poor. Fuzzy logic controllers (FLCs) used in such cases improve the response, but their small-signal response is not as good as that of linear controller. In this paper, design of a SISO-FLC that offers good small and large-signal performance in a boost converter is presented. Besides reducing control complexity, the FLC offers better transient response in the converter than the linear controller. Simulation results are presented to show this. Using describing function method, the system stability margins are also analyzed.

Dynamics and Control of Supercavitating Bodies

Abstract: Analytical and numerical investigations conducted into the control of dive-plane dynamics of supercavitating bodies are presented. Particular attention is paid to tail-slap behavior. A fundamental understanding of the solution structure in terms of equilibrium and other solutions developed through the effort is discussed, and control schemes used to realize stable inner loop dynamics are presented. Dominant nonlinearities associated with planning forces are taken into account in the model and controllability of the system with the fin input and/or the cavitator input has been examined. The describing function method is brought to bear upon this problem and the presence of limit cycles in the controlled and uncontrolled cases are explored. The nonlinear planing force associated with tail-slap behavior is approximated as a piecewise linear function and the results obtained from switching feedback control analysis are provided.Copyright © 2004 by ASME

Rate loop control based on torque compensation in anti-backlash geared servo system

Abstract: This paper proposes the methods that convert two nonlinear characteristics into linear ones in anti-backlash geared servo system. The method to linearize the nonlinear stiffness of the anti-backlash geared train is suggested by investigating the frequency response characteristics. For the purpose of eliminating the nonlinear friction, a new non-model based torque compensator is applied to the servo system. The friction increased by the anti-backlash gear train is measured using torque equation in steady state. The design guideline of this compensator to guarantee the stability is established by describing function analysis and R-H method. In the end, the performance of designed controller is demonstrated by the results of simulation and experiment both in the frequency and in the time domain.

Feedback linearization control of magnetorheological fluid damper based systems with model uncertainty

Abstract: This paper is concerned with the effects of magnetorheological (MR) fluid damper model uncertainties on system stability, when local feedback linearization schemes are utilized to stabilize an inherently unstable system with negative damping. The objectives of the research are to characterize such effects, and develop a robust control law for given uncertainty bounds. First, for the purpose of discussion, sine functions are used as examples to represent the possible deviations between the actual MR damper force and the force predicted by the model. The limit cycle behavior is predicted using the describing function method and parameters influencing the limit cycle characteristics are identified. An effort is then made to identify the worst-case model deviation, given the bounds on the possible model uncertainties. Based on these results and the limit cycle analysis, the minimum control gain to eliminate the limit cycle instability can be derived. With such gains, the closed-loop system is robust against all uncertainties within the bounds.

Gain-phase margin analysis of dynamic fuzzy control systems

Abstract: In this paper, we apply some effective methods, including the gain-phase margin tester, describing function and parameter plane, to predict the limit cycles of dynamic fuzzy control systems with adjustable parameters. Both continuous-time and sampled-data fuzzy control systems are considered. In general, fuzzy control systems are nonlinear. By use of the classical method of describing functions, the dynamic fuzzy controller may be linearized first. According to the stability equations and parameter plane methods, the stability of the equivalent linearized system with adjustable parameters is then analyzed. In addition, a simple approach is also proposed to determine the gain margin and phase margin which limit cycles can occur for robustness. Two examples of continuous-time fuzzy control systems with and without nonlinearity are presented to demonstrate the design procedure. Finally, this approach is also extended to a sampled-data fuzzy control system.

A describing function for resonantly commutated H-bridge inverters

Abstract: The paper presents the derivation of a describing-function to model the dynamic behavior of a metal oxide semiconductor field effect transistor-based, capacitively commutated H-bridge, including a comprehensive explanation of the various stages in the switching cycle. Expressions to model the resulting input current, are also given. The derived model allows the inverter to be accurately modeled within a control system simulation over a number of utility input voltage cycles, without resorting to computationally intensive switching-cycle level, time-domain SPICE simulations. Experimental measurements from a prototype H-bridge inverter employed in an induction heating application, are used to demonstrate a high degree of prediction accuracy over a large variation of load conditions is possible using the simplified model.

Describing function approximation for biomedical engineering applications.

Abstract: This paper focuses on the determination of suitable approximations for sigmoid-type nonlinear characteristics, which are common to physiological systems, particularly cardiovascular regulatory systems. These sigmoid nonlinearities have been implicated in the development of limit cycle oscillations in blood pressure. Approximations of the sigmoid are required since the describing function is not calculable for the all representations of the sigmoid characteristic. In this paper, we present a new approximation, which gives a better overall approximation of the sigmoid and hence, can assist the use of describing functions in the diagnostic analysis of cardiovascular function.

Gain-phase margin analysis of pilot-induced oscillations for limit-cycle prediction

Abstract: This investigation attempts to forecast the limit cycles of pilot-induced oscillations (PIOs) by combining the gain-phase margin tester, the M-locus, and the parameter plane methods. First, one position- or rate-limited nonlinear element is linearized by the conventional means of describing functions. The stability of an equivalent linearized system with adjustable parameters is then analyzed using stability equations and the parameter plane method. Additionally, the minimum gain-phase margin of the PIO system at which a limit cycle can occur is determined by inserting the gain-phase margin tester into the forward open-loop system. Moreover, a simple method is developed to identify the intersections of the M locus and the constant gain and phase boundaries in the parameter plane. In so doing the exact relationship between the gain-phase margin and the characteristics of the limit cycle can be clearly determined. The results of this study demonstrate that these procedures can enhance the analysis of PIO over analysis by other methods in the literature. This approach is extended to PIO analysis with multiple nonlinearities.

Recursive estimation of the Preisach density function for a smart actuator

Abstract: The Preisach operator and its variants have been successfully used in the modeling of a physical system with hysteresis. In an application, one has to determine the density function describing the Preisach operator from measurements. In our earlier work, we described a regularization method to obtain an approximation to the density function with limited measurements. In this paper, we describe methods for recursively computing the approximate density function. These methods can be implemented in real-time controllers for smart actuators.

Vibration Characteristics and Seismic Responses of Mechanical Structures With Hysteresis Elements

Abstract: Hysteresis elements such as elasto-plastic dampers are important for mechanical structures. They have nonlinearities and thus the vibration characteristics of mechanical structures with hysteresis elements are not clarified yet and need further studies. In this paper, the investigation of the dynamical stability of mechanical structures with a hysteresis element is performed by means of describing function method, which is on the basis of linear approximation of the hysteresis element. The nonlinear vibration system is composed of a linear spring-mass-damper structure system of any order and a hysteresis damper. The describing function of a hysteresis damper is derived and it is shown that the function remains upper half plane in the complex plane. According to the passivity of the linear structure system, this fact leads to the vibration system that is stable for any harmonic excitation. Next, the responses for seismic excitation are examined in simulation by using two typical models of the hysteresis element, a bilinear model and a Ramberg-Osgood model. The effects of the second stiffness of the dampers and the amplitude of the excitation motion are investigated using an actually recorded earthquake motion. The simulation results imply that the determination of the second stiffness requires the trade-off between an oscillation and a drift for designing earthquake-proof mechanical structures with the hysteresis damper.

Characterization and Attenuation of Sandwiched Deadband Problem Using Describing Function Analysis and Its Application to Electro-Hydraulic Systems Controlled by Closed-Center Valves

Abstract: Sandwiched deadbands can be seen in a wide variety of systems, such as electro-hydraulic systems controlled by closed-center valves. In such a system, the deadband is between the plant and actuator dynamics and therefore can not be compensated directly like an input deadband. Though this sandwiched deadband problem may be attenuated to certain degree through sophisticated advanced control techniques, the increased cost and the necessity of actuator state feedback prohibit their widespread application in the industry. An economical and popular method is to add an inverse deadband function in the controller to cancel or compensate the highly nonlinear behavior of the deadband. However, such a solution requires that the dynamics before the deadband (eg. the valve dynamics) is fast enough to be neglected — a requirement that can not be met in reality unless the closed loop bandwidth of the overall system is limited very low. To raise the achievable closed loop bandwidth for a much improved control performance, it is essential to be able to precisely characterize the effect of this sandwiched deadband on the stability and performance of the overall closed-loop system, which is the main focus of the paper. Specifically, a describing function based nonlinear analysis will be conducted to predict when the instability will occur and how the resulting limit cycle depends on the actuator dynamics and the targeted closed-loop bandwidth. Based on the analysis, the optimal closed-loop bandwidth can be determined to maximize the achievable overall system performance. The technique is applied to an electro-hydraulic system controlled by closed-center valves to optimize the controller design.Copyright © 2004 by ASME

Two measurement techniques to determine Higher Order Sinusoidal Input Describing Functions

Abstract: For high precision motion systems, modelling and control design specifically oriented at friction effects is instrumental. The Sinusoidal Input Describing Function theory represents a solid mathematical framework for analysing non-linear system behaviour. This theory however limits the description of the non-linear system behaviour to an approximated linear relation between sinusoidal excitation and sinusoidal response. An extension to Higher Order Describing Functions can be realised by calculating the corresponding Fourier coefficients. The resulting Higher Order Sinusoidal Input Describing Functions (HOSIDFs) relate the magnitude and phase of the higher harmonics of the periodic system response to the magnitude and phase of a sinusoidal excitation. This paper describes two techniques to measure HOSIDFs. The first technique is FFT based. The second technique is based on IQ (=in phase/quadrature phase) demodulation. In a case study both techniques are used to measure the changes in dynamics due to friction as function of drive level in an electric motor.

Analysis of steady state behavior of second order sliding mode algorithms

Abstract: An analysis of a second order sliding mode algorithm, which is known as the super-twisting algorithm, 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 comparison between the periodic solutions in the systems with higher order sliding mode controllers and the oscillations that occur in classical sliding mode systems with actuators is done.

Convergence and computation of describing functions using a recursive Volterra series

Abstract: Presented in this paper is a comparison of algorithms for computing an approximation to the sinusoidal input describing function (SIDF) for the nonlinear differential equation ẏ(t)+b1y(t)+b2u2(t)y(t) = K(u(t)+b3u(t)) The importance of this nonlinear differential equation comes from the context of nonlinear feedback controller design Specifically, this equation is either a linear lead or lag controller (depending on the coefficient values) augmented with a nonlinear, polynomial type term Consequently, obtaining a SIDF representation of this nonlinear differential equation or creating a process to obtain SIDFs for other similar differential equations, will facilitate nonlinear controller design using classical loop shaping tools The two SIDF approximations studied include the well-established harmonic balance method and a Volterra series based algorithm In applying the Volterra series, several theoretical issues were addressed including the development of a recursive solution that calculates high order Volterra transfer functions, and the guarantee of convergence to an arbitrary accuracy Throughout the paper, case studies are presented Copyright © 2004 John Wiley & Sons, Ltd

A Gain Scheduling Controller for Friction Compensation

Abstract: Friction in mechanical systems can show a very complex behaviour. In addition to its non-linear dependence on the velocity, friction is also function of the position. In the vicinity of motion reversal points, i.e. in the prerolling/pre-sliding regime, the position dependence dominates and appears in the form of a rate-independent hysteresis with nonlocal memory. One of the ways of treating this complex behaviour is through the use of the Describing Function (D.F.) method. This approach yields equivalent dynamic parameters, which are function of the amplitude of motion.

Nonlinear devices characterization and micromachining techniques for RF integrated circuits

Abstract: The present work is dedicated to the development of high performance integrated circuits for wireless communications, by acting of three different levels: technologies, devices, and circuits. Silicon-on-Insulator (SOI) CMOS technology is used in the frame of this work. Micromachining technologies are also investigated for the fabrication of three-dimensional tunable capacitors. The reliability of micromachined thin-film devices is improved by the coating of silanes in both liquid- and vapor-phases. Since in telecommunication applications, distortion is responsible for the generation of spurious frequency bands, the linearity behavior of different SOI transistors is analyzed. The validity range of the existing low-frequency nonlinear characterization methods is discussed. New simple techniques valid at both low- and high-frequencies, are provided, based on the integral function method and on the Volterra series. Finally, the design of a crucial nonlinear circuit, the voltage-controlled oscillator, is introduced. The describing function formalism is used to evaluate the oscillation amplitude and is embedded in a design methodology. The frequency tuning by SOI varactors is analyzed in both small- and large-signal regimes.

Pilot-Induced Oscillation Analysis with Actuator Rate Limiting and Feedback Control Loop

Abstract: Fully-developed pilot-induced oscillation (PIO) is an important issue to be solved in the development of modern fly-by-wire flight control systems. In this paper, the fully-developed PIO is analyzed as a worst case for the safety of piloted airplanes, including actuator rate limiting, feedback control loop, and pilot delay by using describing function method. It is shown that the predictions obtained with this method closely match results of the simulation in the frequency and the amplitude of the PIO limit cycle. And it demonstrates that the feedback control loop has a positive effect on PIO and decreases amplitude of the oscillation.

Describing Functions For Effective Stiffness and Effective Damping of Hystersis Structures

Abstract: For a hysteresis structure with energy dissipation devices, the force-displacement relation is nonlinear such that it is very difficult to evaluate the actual damping and stiffness coefficients, even if the forcedisplacement characteristic is simply perfect elasto-plastic. With the describing function method, we can linearize the nonlinear behavior of the energy dissipation devices and then obtain the equivalent damping and stiffness coefficients; In turn, the effective period and equivalent damping ratio can be attained. It is stressed that with this approach, the imaginary part of the describing function is just the energy dissipation term, which corresponds to the conventional hystersis damping derived by the energy method. Simulation results confirm the effectiveness of this proposed method.

Limit cycle oscillation and domain of stability in prototypical wing sections with unsteady aerodynamics

Abstract: The paper treats the question of the existence of limit cycle oscillations and domain of stability (attraction) of prototypical aeroelastic wing sections with pitch structural nonlinearity using the describing function method. The model includes unsteady aerodynamics based on Theodorsen’s theory. The dual-input describing functions of the nonlinearity are used for the limit cycle analysis. Analytical expressions for the computation of the average value, and the amplitude and frequency of oscillation of pitch and plunge responses are obtained. Interestingly, it is found that flutter can exist not only when the origin in the state space is unstable but also when it is asymptotically stable if the initial conditions are not small. For such cases, an estimate of the domain of stability surrounding the origin in the state space is computed in which flutter cannot exist. The Nyquist criterion is used to establish the stability of the limit cycle and it is shown that unstable as well as the stable limit cycles exist when the origin is exponentially stable. Numerical results are presented for a set of values of the flow velocities and the locations of the elastic axis which show that the predicted limit cycle oscillation amplitude and frequency as well as the mean value are quite close to the actual values.