Showing papers on "Describing function published in 2003"

Precision motion control with disturbance observer for pulsewidth-modulated-driven permanent-magnet linear motors

Abstract: In this paper, we address the problem of precision motion control of permanent-magnet linear motors (PMLMs) under the influence of significant disturbances. We establish a mathematical model of a PMLM driven by a sinusoidal pulsewidth-modulated (PWM) amplifier, obtaining it from a describing function analysis of the essentially nonlinear characteristics. The overall model (PWM+PMLM) inevitably inherits uncertainties in the face of load changes, system parameter perturbation, noise, and inherent system nonlinearities, etc., all of which constitute disturbances to the control system that will adversely affect the precision and accuracy. We propose a robust control scheme employing a disturbance observer to address the sensitivity of the control performance to the disturbances. Real-time experimental results are provided to verify and confirm the practical effectiveness of the proposed approach.

Nonlinear Control Systems

Abstract: Properties of nonlinear systems stability linearization methods operating modes and dynamic analysis methods phase trajectories in dynamic analysis of nonlinear systems harmonic linearization in dynamic analysis of nonlinear control systems operating in stabilization mode harmonic linearization in dynamic analysis of nonlinear control systems in tracking mode of operation performance estimation of nonlinear control system transient responses describing function method in fuzzy control systems. Appendices: harmonic linearization Popov diagrams.

An H-infinity-based control design methodology dedicated to the active control of vehicle longitudinal oscillations

Abstract: This paper deals with the problem of active damping of vehicle oscillations. A complete methodology based on an H/spl infin/ optimization is presented. Several dedicated analysis tools are used in order to analyze the behavior of the closed-loop system. Two tuning parameters allow the design of a controller managing the compromise between performance (in terms of oscillation attenuation and limit cycles) and robustness. Finally, a feedforward component is designed in order to improve the performance without changing the robustness properties or the controller complexity. Simulation results obtained with an experimentally validated model show the efficiency of the resulting controller.

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 with the use of the describing function and the locus of a perturbed relay system approach. 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 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.

Nonlinear Dynamic Modelling of Gear-Shaft-Disk-Bearing Systems Using Finite Elements and Describing Functions

Abstract: This study presents a new nonlinear dynamic model for a gear-shaft-disk-bearing system. A nonlinear dynamic model of a spur gear pair is coupled with linear finite element models of shafts carrying them, and with discrete models of bearings and disks. The nonlinear elasticity term resulting from backlash is expressed by a describing function, and a method developed in previous studies to determine multi harmonic responses of nonlinear multi-degree-of-freedom systems is employed for the solution. The excitations considered in the model are external static torque and internal excitation caused by mesh stiffness variation, gear errors and gear tooth profile modifications. The model suggested and the solution method presented combine the versatility of modeling a shaft-bearing-disk system that can have any configuration without a limitation to the total degree of freedom, with the accuracy of a nonlinear gear mesh interface model that allows to predict jumps and double solutions in frequency response. Thus any single stage gear mesh configuration can be modeled easily and accurately. With the model developed it is possible to calculate dynamic gear loads, dynamic bearing forces, dynamic transmission error and bearing displacements. Theoretical results obtained by using the method suggested are compared with the experimental data available in literature, as well as with the theoretical values calculated by employing a previously developed nonlinear single degree of freedom model.

New techniques for non-linear behavioral modeling of microwave/RF ICs from simulation and nonlinear microwave measurements

Abstract: This paper compares and contrasts recent nonlinear behavioral modeling techniques designed for microwave and RFIC applications which arise in radio and communication systems, and in the design of broad-band nonlinear components used for microwave instrumentation. These techniques include dynamic neural networks and nonlinear time series models in the time-domain, nonlinear describing functions in the frequency domain, and envelope-based methods in mixed time and frequency domains. Approaches to generating these models from both simulation and nonlinear microwave measurements are reviewed.

Development and test of MEMS accelerometer with self-sustatined oscillation loop

Abstract: In this paper, presented are summarized results of the design, analysis, and experiment of a self-sustained oscillation loop for a surface micromachined resonant accelerometer, ACRC-RXL. After introduction of the operating principle, the micromachined device is illustrated including the fabrication process. In designing the oscillation loop, a describing function technique based on the nonlinear system analysis approach is applied. Using the analytic results, a particular loop parameter is designed and loop performance is characterized. Then a real accelerometer system is implemented using mechanical structure and analog electronics. Experimental results show that the designed oscillation loop works well in real environments and the closed loop system has good performances of bias stability about 0.7 mg and dynamic range over 10 g.

Limit Cycle Oscillation and Orbital Stability in Aeroelastic Systems with Torsional Nonlinearity

Abstract: The paper treats the question of the existence of limit cycleoscillations of prototypical aeroelastic wing sections with structuralnonlinearity using the describing function method. The chosen dynamicmodel describes the nonlinear plunge and pitch motion of a wing. Themodel includes an asymmetric structural nonlinearity in the pitchdegree-of-freedom. The dual-input describing functions of thenonlinearity are derived for the limit cycle analysis. Analyticalexpressions for the average value, and the amplitude and frequency ofoscillation of pitch and plunge responses are obtained. Based on ananalytical approach as well as the Nyquist criterion, stability of thelimit cycles is examined. Numerical results are presented for a set ofvalues of the flow velocities and the locations of the elastic axiswhich show that the predicted limit cycle oscillation amplitude andfrequency as well as the mean value are quite close to the actualvalues. Furthermore, for the chosen model with linear aerodynamics, itis seen that the amplitude of the pitch limit cycle oscillation does notalways increase with the flow velocity for certain elastic axislocations.

An LQG guidance law with bounded acceleration command

Abstract: This paper presents a novel missile guidance law derived by analyzing an interception scenario in the framework of an LQG terminal control problem with a bound on the acceleration command. For the derivation, the target maneuver is represented by an appropriate shaping filter and the nonlinear saturation function is represented by the equivalent random input describing function. Since the certainty equivalence property is not valid in the investigated problem the resulting controller depends on the conditional probability density function of the estimated states. The new guidance law has the same structure as the classical optimal guidance law that was derived without taking into account the saturation in the control. However the navigation gain, computed by solving numerically a TPBVP, has unique characteristics such as not diverging near the terminal time. Using Monte Carlo simulations it is shown that the homing performance of the new guidance law is better than that of the classical one; moreover, the control effort is drastically reduced. The results validate the new derivation approach.

Nonlinear Response of Aeroservoelastic Systems Using Discrete State-Space Approach

Abstract: A generalized direct simulation method using a discrete time-domain state-space approach for transient response of both open- and closed-loop nonlinear aeroelastic systems is developed. Based on a nonlinear parameter scheme that divides the nonlinear system into sublinear systems, the method first assembles a set of discrete time-domain state-space equations and then computes the transient response by switching the time-integration procedure between this set of state-space equations. In so doing, various nonlinearities in structures, aerodynamics and/or control systems can be included. The method is validated by correlating the transient response of a three-degree-of-freedom airfoil section in freeplay with the experimental and numerical results obtained by Conner et al. The stability of a strut-braced wing with buckling effects at two trim conditions is also studied, which shows that the aeroelastic stability of the present strut-braced wing is trim-condition dependent. Such results clearly could not be obtained if using a linear aeroelastic analysis.

A view on limit cycle bifurcations in relay feedback systems

Abstract: The paper is concerned with the study of oscillations in linear dynamic systems with relay feedback. The specific interest is about the bifurcations of these periodic solutions, with regard to phenomena which also occur in smooth systems and to others due to the relay discontinuity. The followed approach moves from the describing function method, leading to results which are approximate in nature but which express in a simple and correct way the essential mechanisms of the studied phenomena, as shown in the proposed examples.

Automatic computation of polyharmonic balance equations for non-linear differential systems

Abstract: An automated algorithm is presented which enables the harmonic balance equations for any polynomial type pure or cross-product non-linear differential system 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, in the form of a multi-input, amplitude dependent describing function, is therefore readily computed. The method is illustrated by means of an example, and the results validated against detailed numeric simulation.

Reliable and accurate algorithm to compute the limit cycle locus for uncertain nonlinear systems

Abstract: A reliable and accurate algorithm is proposed to compute the so-called limit cycle locus for separable nonlinear systems with nonlinear parametric uncertainty. The limit cycle locus for a given uncertain parameter is the locus of the limit cycle points as the parameter varies over its range. The uncertain parameter may be associated with either the linear or nonlinear element in the system. The proposed algorithm makes use of the describing function analysis technique and tools of interval analysis for predicting the limit cycle behaviour. The capability of the proposed algorithm is demonstrated on an example that can not be readily solved using existing methods. An additional example shows how the proposed algorithm can also be applied to tune a controller to achieve a prescribed limit cycle behaviour.

New approach to assessing the effects of parametric variations in feedback loops

Abstract: A method for the computation of the magnitude and phase envelopes of uncertain transfer functions is presented. The idea is to factor the transfer function into its real and complex pair roots and find the maximum and minimum magnitudes of the gain and phase of each factor. The Bode envelopes of the given uncertain system are then found from those of the individual factors. This approach, which is different from those based on the interval polynomial method of Kharitonov, has the major advantage that the representation is more applicable to practical situations where typically the coefficients of the various factored terms relate to physical parameters of a mathematical model. Further, the method results in the narrowest Bode envelopes and therefore can yield improved controller designs. The describing function analysis and absolute stability problem of nonlinear systems with variable plant parameters are also studied. An approach which enables one to predict the existence of limit cycles in a control system which simultaneously contains nonlinearities and parametric uncertainties is given. The proposed method makes use of the popular describing function technique and these narrowest possible Bode envelopes of linear uncertain transfer functions in factored form. The technique can be used to cover the cases of linear elements, which have a multilinear or nonlinear uncertainty structure, and a nonlinear element with or without memory. New formulations of the circle and off-axis circle criteria are given for use with Bode diagrams so that the absolute stability of nonlinear systems with variable plant parameters can be studied. Examples are given to show how the proposed method can be used to assess the effects of parametric variations in feedback loops.

Time-Domain Analysis of Nonlinear StressGrading System for High Voltage Rotating Machines

Abstract: This paper introduces a new approach for the design and analysis of nonlinear stress-grading systems for high voltage rotating machines. The method is based on the simulation and modeling of sues-grading systems using a nonlinear lumped circuit model. The nonlinear model is analyzed in the time domain using the describing function method. The model generates the time variation of the resistances as well as the surface electric field and potential and ultimately converges to the optimal design parameters of the stress-grading system. The model is iterative and adaptive where a deviation from the nominal uniform field and the upper and lower bounds on the resistance values as well as bunds on the nonlinearity factor can be set to generate optimal design parameters of the stress- grading system

Time-domain analysis of nonlinear stress-grading systems for high voltage rotating machines

Abstract: This paper introduces a new approach for the design and analysis of nonlinear stress-grading systems for high voltage rotating machines. The method is based on the simulation and modeling of stress-grading systems using a nonlinear lumped circuit model. The nonlinear model is analyzed in the time domain using the describing function method. The model generates the time variation of the resistances as well as the surface electric field and potential and ultimately converges to the optimal design parameters of the stress-grading system. The model is iterative and adaptive where a deviation from the nominal uniform field and the upper and lower bounds on the resistance values as well as bounds on the nonlinearity factor can be set to generate optimal design parameters of the stress-grading system.

Infinity norm measurement of sensitivity function based on limit cycles in a closed-loop experiment

Abstract: This paper proposes a new method for the measurement of the infinity norm of the sensitivity functions of linear systems. A nonlinear feedback structure is derived from existing relay experiments. The proposed scheme may produce a limit cycle at a frequency where the sensitivity function of a given system achieves a selected magnitude. Necessary and sufficient conditions for the occurrence of the limit cycles are discussed using the describing function method. If the magnitude is chosen larger than the sensitivity infinity norm, the strictly passivity of the scheme can be demonstrated. Based on these analyses, a simple procedure is then proposed to identify the infinity norm of the sensitivity function for a large class of processes. Simulation results show the effectiveness and simplicity of the proposed method.

Avoidance of nonlinear resonance jump in turbine governor positioning system using fuzzy controller

Abstract: The paper deals with analysis of turbine governor positioning system of hydroelectric power plant regarding nonlinear resonance jump. The term "resonance jump" is used in the case of a sudden jump of amplitude and/or frequency and/or phase of a periodic output signal of a nonlinear system. Resonance jump cannot occur if excitation is such that the response of the system is transient and cannot be defined by solving nonlinear differential equations. Resonance jumps can cause severe damage and the mechanical, hydraulic and electrical systems with the saturation should be analyzed with the respect to the possibility of occurrence of resonance jump. In the paper the analysis of the water turbine governor positioning system with the respect to the occurrence of resonance jump is described. Two methods of analyzing system with the respect to resonance jumps are described: the simulation method and the analytical method. When using the simulation method, the analysis of the variations of important parameters, including dead zone is performed and the simulation procedures are detailed. Then, the confirmation of the results of simulation is given by the analytical analysis method based on the describing function approach. At the end, the fuzzy regulator for avoidance of nonlinear resonance jump in turbine governor positioning system is involved.

An algebraic approach to the design of robust limit cycle controllers

Abstract: The design of robust limit cycle controllers introduced here can be used for autonomous systems with separable single-input-single-output nonlinearities and unavoidable limit cycles. The objective is to design a controller to secure specified oscillation amplitude and frequency. The method consists of quasi-linearization of the nonlinear element via a describing function (DF) approach and then shaping the loop to reach desired limit cycle characteristics. As the DF method is used, loop shaping takes place in the Nyquist plot.

A new approach to the stability of nonlinear systems with uncertain plant parameters

Abstract: The purpose of this paper is to study the problem of the stability of nonlinear systems with variable plant parameters. A new approach which enables one to predict the existence of limit cycles in a control system which simultaneously contains nonlinearities and parametric uncertainties is given. The proposed method makes use of the popular describing function technique and the Bode envelope of linear uncertain transfer functions. The narrowest possible Bode envelopes are obtained from new results obtained by the authors where the plant transfer function is taken in factored form. The technique can be used to cover the cases of linear elements, which have multilinear or nonlinear uncertainty structure, and a nonlinear element with or without memory. Examples are given to show how the proposed method can be used to assess the stability of nonlinear systems with uncertain plant parameters.

Limit cycle analysis of PID controller design

Abstract: The main purpose of this paper is to predict limit cycles of nonlinear PID control systems with adjustable parameters by the approaches of stability equation, describing function and parameter plane. First, nonlinear elements are linearized by the classical describing function method. The stability of equivalent linearized system with adjustable parameters is then analyzed by using the stability equations and the parameter plane method. According to our study, the characteristics of limit cycles with PID control design in current literature can be improved. Finally, this approach is also extended to a sampled-data control system with broken line nonlinearity.

Systems with Commensurate Delays

Abstract: Stability criteria based on a frequency domain representation are time-honored tools in the study of dynamical systems. Classical examples of frequency domain stability criteria include such results as the Nyquist test and the root-locus method. With the aid of the small gain theorem, frequency domain tests have become increasingly more prevalent in stability analysis, and have played an especially central role in the theory of robust control. More generally, while frequency domain methods are used predominantly in the analysis of linear systems, they have also found utilities in the study of nonlinear systems, with such tools as describing functions, Popov and circle criterion, and the small gain theorem as well. Various frequency-sweeping tests are now commonplace. Generally, frequency domain tests are often favored for their conceptual simplicity and computational ease, which typically can be checked in an efficient manner by plotting graphically a certain frequency-dependent measure.

Novel techniques for optimal design and analysis of corona-suppression systems

Abstract: The use of stress-grading systems proved to be essential to suppress corona in the end-turn zone of high voltage machines Moreover, power dissipation in the end-turn region is believed to be a major factor in the deterioration process of surface field grading systems This paper introduces two novel techniques for the design and analysis of both linear and nonlinear corona-suppression systems of high voltage machines The first technique is based on the design of linear stress-grading systems through power-loss minimization and equalization of electric field along the end-turn zone The second technique is based on the simulation, modeling and analysis of nonlinear stress-grading systems in the time domain using the describing function method This model generates the time variation of the resistances as well as the surface electric field and potential and ultimately converges to the optimal design parameters of the stress-grading system

Performance Analysis of the Binary Flow Control Algorithm

Abstract: ABR (available bit rate) flow control is an effective measure in ATM network congestion control and traffic management. In large scale and high-speed network, the simplicity of the algorithm is crucial to optimize the switch performance. Though the simplicity of binary flow control is very attractive, the queue length and allowed cell rate (ACR) controlled by the standard EFCI algorithm oscillate with great amplitude, which has negative impact on the performance, so its applicability is doubted, and then relatively complex but effective explicit rate feedback algorithms are introduced and explored. In this study, based on the existed flow control model, the performance of standard EFCI algorithm is evaluated and analyzed with the describing function approach in nonlinear control theory, concluding that queue and cell rate self-oscillations are caused by the inappropriate nonlinear control law originated from intuition, but not intrinsic attribute of the binary flow control mechanism. The simulation experimental results are done to validate this analysis and conclusion. Finally, a parameter settings scheme is put forward to optimize the existed EFCI switch.

Learning complex systems from data: the Set Membership approach

Abstract: In the paper the problem of making inferences on unknown nonlinear system (e.g. identification, prediction, smoothing, filtering, control design, decision making, fault detection, etc.) based on finite and noise-corrupted measurements is considered. Inferences are usually obtained by means of models of the system, estimated from measurements within a finitely parametrized model class describing the functional form of involved nonlinearities, whose proper choice is realized by a search, from the simplest to more complex ones (linear, bilinear, polynomial, neural networks, etc.). In this paper an alternative approach, recently developed by the authors is presented. The approach, based on a Set Membership framework, does not need assumptions on the functional form of the regression function describing the system, but requires only some information on its regularity, given by bounds on the derivatives. In this way, the problem of considering approximate functional forms is circumvented. Moreover, noise is assumed to be bounded, in contrast with statistical methods, which rely on assumptions such as stationarity, ergodicity, uncorrelation, type of distribution, etc., whose validity may be difficult to be reliably tested and is lost in presence of approximate modeling. In this paper some of the main results developed by the authors are presented within a unifying framework. In particular, necessary and sufficient conditions for checking the assumptions validity are given and optimal and almost optimal algorithms are presented for the cases that the desired inferences are identification and prediction.

The describing function method accuracy in first order plants with rate-limited feedback

Abstract: In this paper, the rate limiter describing function is used to analyze by harmonic balance the existence of limit cycles in first order plants. Next, by comparison with analytical results previously obtained by the authors, the accuracy of the above approximate predictions is assessed. The main conclusion is that from a qualitative point of view the harmonic balance is good enough, providing in a exact way the critical value of the bifurcation parameter, and leading to neither spurious predictions of limit cycles nor failures in their detection. Regarding the quantitative aspects, the accuracy of the method depends on certain dimensionless parameter which is intrinsic to the problem.

Analysis and synthesis of robust control system with separable nonlinearities.

Abstract: The objective of this thesis is to propose a design method for controllers to secure specified robust oscillation amplitude and frequency for separable nonlinear systems exhibiting unavoidable or desirable limit cycles. The design approach of robust limit cycle controllers introduced here can be used for autonomous systems with separable single-input-single-output nonlinearities. The proposed design approach consists of quasi-linearization of the nonlinearity via the Describing Function (DF) method and then shaping the loop to reach desired limit cycle characteristics. As the DF method is used, loop shaping takes place in the Nyquist plot, because the intersection between the loop shaped loci and the DF negative reciprocal loci is related to the limit cycle characteristics of the system. In the problem addressed in this thesis, the original linear subsystem (i.e. without controller) is considered as an uncertain system with unstructured uncertainty. A condition to be satisfied by the shaped loop is given such that the controlled system has limit cycle characteristics as close as possible to the nominal characteristics even though the uncertainty considered. Three requirements are satisfied in the loop shaping step. The first requirement assures the nominal desired limit cycle characteristics in the designed system. The second is related to limit cycle stability, i.e., if a (small) perturbation disturbs the limit cycle of the controlled system, it will return to the earlier limit cycle. The third requirement addresses the robustness question, and the condition mentioned before is used. After the loop is shaped to satisfy the requirements, a controller is computed so that the controlled linear subsystem transfer function is equal the transfer function obtained in the loop shaping. This approach is applicable to systems with linear subsystems that are minimum phase. In some cases it may be necessary to add a high frequency pole in order to obtain a proper controller. Examples are given in order to illustrate the robustness of the controlled system with respect to uncertainty in the linear subsystem model. In the first example the intersection between the loop shaped loci and the DF negative reciprocal loci is on the real axis. In the second example the crossing point is not on the real axis and so, restrictions more generals than the ones used in the first example are used. In the third example the crossing point is on the real axis and output disturbance is considered.