Showing papers on "Describing function published in 1992"

Small-signal modeling of series and parallel resonant converters

Abstract: A small-signal modeling technique based on the extended describing function concept is applied to series-resonant converters (SRCs) and parallel-resonant converters (PRCs). The models developed include both frequency control and phase-shift control. The small-signal equivalent circuit models are also derived and implemented in PSPICE. The models are in good agreement with measurement data. The high-frequency dynamics of resonant converters around the beat frequency can be accurately modeled. These simple analytical models can be employed in the control loop design of resonant converters. >

Small-signal modeling of LCC resonant converter

Abstract: Small-signal modeling based on the extended describing function concept is applied to LCC resonant converters. The analytical model developed includes both frequency and phase-shift control. Small-signal equivalent circuit models are also derived and implemented in PSPICE. The models ae in good agreement with the measurement data. >

A harmonic balance approach for chaos prediction: chua’s circuit

Abstract: The paper proposes a practical engineering approach for predicting chaotic dynamics in an important class of nonlinear systems. The aim of this approach is to provide a heuristic method of analysis which can give reasonably accurate answers but is far simpler to apply than other more rigorous methods based on nonlinear dynamics. Our approach is founded on the harmonic balance principle and uses standard describing function techniques well known to design engineers. Our method consists of a synergism of two independent techniques, each one constituting a possible mechanism for chaos. These two techniques are combined into a single algorithm which is highly efficient computationally, taking only a fraction of the time normally required by other more exact procedures. In order to present and illustrate our algorithm clearly, and in order to compare its predictions with readily available results obtained by rigorous methods, we have chosen Chua’s circuit as a vehicle to demonstrate the effectiveness of this approach. Chua’s circuit was chosen not only for the huge amount of results already published concerning the dynamics of this system, but also because it represents a real physical system easily built in the laboratory, and whose simple mathematical model has proven to be realistic and mathematically tractable. Although our algorithm does not guarantee its prediction is fully reliable, any more than the widely used describing function method does, its significance is based entirely on the empirical evidence that it yields qualitatively correct, though not exact, results for all of the chaotic phenomena that we have investigated in Chua’s circuit, as well as in many other chaotic systems. We hope therefore that our chaos prediction algorithm will find practical uses among engineers and scientists not familiar with more specialized mathematical approaches, in search of a simple and practical, although not rigorous, tool for analyzing systems with complex dynamics.

Describing functions and limit cycles

Abstract: The conditions for limit cycling of a single-loop nonlinear feedback system are considered in terms of the describing function for a single-valued symmetrical time- and frequency-independent nonlinearity. Additional assumptions on the polynomial are required. >

Nonlinear system dynamics

Abstract: Linearity-A Shangri-La.- I.1 Why Linearity?.- I.2 From Whence Linearity.- I.3 What is Linearity?.- I.4 Things Linear or Not Linear.- I.5 Our Shangri-La.- 1 Four Interesting Equations.- 1.0 The Equations.- 1.1 The Solutions-A Preview.- 1.2 Solving Equation (i): ? + x = 0.- 1.3 Solving Equation (ii): ? + ?x? = 0.- 1.4 Solving Equation (iii): ??? + x = 0.- 1.5 Solving Equation (iv): ??? + ??? = 0.- 1.6 A Variation on Equation (iv): ??? = ???.- 1.7 Existence/Uniqueness of Solutions.- 1.8 Problems.- 1.9 References and Related Literature.- 2 Analytic Solutions to Nonlinear Differential Equations.- 2.0 Introduction.- 2.1 First-Order Base Equations.- 2.2 Second-Order Base Equations.- 2.3 The Ricatti Equation.- 2.4 Nonlinear Base Equations.- 2.5 Derivative and Integral Functional Relations.- 2.6 Matrix Nonlinear Differential Equations.- 2.7 Application to the Calculus of Variations.- 2.8 Problems.- 2.9 References and Related Literature.- 3 Linearization.- 3.0 Introduction.- 3.1 Linearizing Algebraic Functions.- 3.2 Linearizing a Transistor.- 3.3 Linearization of Differential Functions.- 3.4 Linearizing Satellite Motion.- 3.5 Concluding Example.- 3.6 Summary.- 3.7 Problems.- 3.8 References and Related Literature.- 4 The Describing Function.- 4.1 Describing Function.- 4.2 Frequency-Dependent Describing Functions.- 4.3 Digital Simulation Verifies Describing Function Analysis.- 4.4 Asymmetric Describing Functions.- 4.5 Problems.- 4.6 References.- 5 Some Properties of Nonlinear Systems.- 5.0 Introduction.- 5.1 Linear System Characteristics.- 5.2 Nonlinear Equations with Periodic Solutions.- 5.3 Limit Cycles.- 5.4 Nonlinear System Behavior.- 5.5 Some Physically Realizable Nonlinearities.- 5.6 Problems.- 5.7 References and Related Literature.- 6 Liapunov Stability.- 6.0 Introduction.- 6.1 Liapunov Stability: An Overview.- 6.2 Construction of Liapunov Functionals.- 6.3 The Lur'e Problem.- 6.4 The Popov Criterion.- 6.5 Problems.- 6.6 References and Related Literature.- 7 Recursions and their Stability.- 7.1 Recursions.- 7.2 The Mechanic's Rule.- 7.3 Singularities and Peculiarities.- 7.4 The Logistics Map.- 7.5 Recursing Differential Equations.- 7.6 Problems.- 7.7 References.- 8 Digital Simulation.- 8.0 Background.- 8.1 Recursion Formulae: Fundamental to Digital Simulation.- 8.2 The Sampling Process Creates Discrete Data.- 8.3 An Approach to Digital Simulation: Introduce Samplers.- 8.4 Concept of Pulse Filters Implicit in Digital Simulations.- 8.5 Introducing Z-Transforms.- 8.6 The Pulse Filter Now Becomes a Z-Transform.- 8.7 Introductory Example of Digital Simulation.- 8.8 Digitally Simulating a Feedback System.- 8.9 Simulating Hysteresis Due to Backlash in Gears.- 8.10 Simulating a System with Hysteresis.- 8.11 Problems.- 8.12 References.- 9 Spreadsheet Simulation-A Tutorial.- 9.1 Introduction.- 9.2 Use of the "Copy" Command.- 9.3 Using the Graph Command.- 9.4 Summary.- 9.5 Some Heuristic Exercises-Brief Descriptions.- 9.6 Suggested Solutions to the Exercises.- 9.7 References.- 10 An Isobaric Cabin Pressure Control.- 10.0 Introduction.- 10.1 Background.- 10.2 How the Cabin is Pressurized.- 10.3 Basic Numbers and Constraints.- 10.4 Cabin Dynamics.- 10.5 Design Evolution.- 10.6 A Nonlinear Digital Simulation.- 10.7 Limiting Cabin Pressure Rate.- 10.8 Initializing Integrators.- 10.9 Problems.- 10.10 References and Related Literature.

Coupled modal sliding mode control of vibration in flexible structures

Abstract: The sliding mode control algorithm is developed for the coupled modal space control of a flexible structure when the bounds on the system parameters' errors are known. An explicit method to construct the desired sliding hyperplanes for the coupled modal sliding mode control is formulated. A boundary layer is used around each sliding hyperplane to eliminate the chattering phenomenon. Three types of steady-state solutions for the closed-loop system inside the boundary layers are found: the zero solution (origin of the state space), the constant nonzero solution, and the limit cycle. The amplitudes, phase angles, and the frequency of the limit cycle have been estimated by the describing function approach. The modal displacements corresponding to the constant nonzero solution have been obtained analytically. The stability of the zero solution has been examined by the linearized system analysis. Using a flexible tetrahedral truss structure, numerical examples are presented to verify the theoretical analyses.

H/sub infinity / control of nonlinear systems using describing functions and simplicial algorithms

Abstract: A novel method for generating the optimal H/sub infinity / controller for a nonlinear system is discussed. It is a three-step approach encompassing modeling (using the describing function approach), synthesis (using the H/sub infinity / loop shifting technique), and robustness analysis (using the labeling technique of simplicial algorithms). The method is first demonstrated by designing an optimal H/sub infinity / controller for an unstable system driven by a bang-bang actuator. This system was implemented on a digital computer and in analog circuit form to demonstrate the practicality of the method. >

Design of Sampled-Data Control Systems with One Memoryless, Time-Invariant Nonlinearity using QFT Technique

Abstract: In this paper, we examine the controller design for sampled-data systems containing one memoryless, time-invariant nonlinearity within the plant. The linear part of the plant is subjected to large parameter variations. By using Kochenburger's describing function analysis for the nonlinear part, modified templates of the plant at a set of selected frequency points can be constructed on the Nichols chart. The quantitative feedback theory (QFT) is then used to design the robust controllers. Two examples with saturation and backlash types of nonlinearity are given to demonstrate the effectiveness of this approach.

Output feedback controller for systems containing a backlash

Abstract: For the case of a single-input single-output control system with a backlash, the feedforward gain is determined. A control equation is applied to the original system. Because of the nonlinearity, the resulting feedback system often exhibits a limit cycle. In this case, the feedback gains are readjusted using the describing function method so that the limit cycle is eliminated. The block diagram for describing function analysis is given. It is shown that, when the controller is implemented, the steady state error is related to the sum k/sub 0/+k/sub 1/+k/sub 2/+. . .+k/sub n-1/ of the feedback gains. Although the relationship is not linear, the larger the sum, the smaller the error becomes. In addition to satisfying the transient requirements, it is also necessary to choose the cost weighting matrices Q and R so that the sum of the feedback gains is sufficiently large and the steady state error is acceptably small. >

Describing Function-based Analysis of a Nonlinear Hydraulic Transmission Line

Abstract: This paper presents an analytical method of predicting and eliminating limit cycle conditions in hydraulic drive systems, where backlash in the actuator seal is the dominant nonlinearity. It is shown that limit cycle conditions are a strong function of certain plant parameters. The elimination of limit cycle conditions may thus be achieved by the alteration of certain inherent plant parameters. Fluid flow in a hydraulic pipe is modelled using transmission line theory. The sinusoidal-input describing function is used to indicate which values of the plant parameters would result in limit cycle conditions. The analytical results were verified on an experimental test bench.

A method for closed loop automatic tuning of PID controllers

Abstract: A simple method for the automatic tuning of PID controllers in closed loop is proposed. A limit cycle is generated through a nonlinear feedback path from the process output to the controller reference signal. The frequency of this oscillation is above the crossover frequency and below the critical frequency of the loop transfer function. The amplitude and frequency of the oscillation are estimated and the control parameters are adjusted iteratively such that the closed loop transfer function from the controller reference to the process output attains a specified amplitude at the oscillation frequency.

Application of hopf bifurcation at infinity to hunting vibrations of rail vehicle trucks

Abstract: SUMMARY Many models of train wheelset systems involve Coulomb friction terms and model the flange rail forces as stiff linear springs with dead bands. For such systems, the regular theory of Hopf bifurcation is not applicable, whereas a newly developed theory of Hopf bifurcation at infinity can be used. This is because such systems are basically linear as far as very large amplitude motions are concerned. In this paper, we apply this theory to study the oscillations of a wheelset using a numerical algorithm. Comparisons with the results obtained by the describing function method are also made.

Justification of the describing function method for periodically switched circuits

Abstract: The author studies the describing function method for analysis of periodically switched circuits, such as those used in power electronic applications. Typical power circuit models have nonlinear elements that do not satisfy a Lipschitz continuity condition. As a result, classical averaging theory is not generally applicable to these switched systems. Furthermore, as a result of the nonsmooth characteristics of circuit nonlinearities, previously developed justifications for the describing function method are also not applicable. A justification is developed for the describing function method that relies on the incrementally passive characteristics of the network elements comprising typical power electronic circuits. The analysis generates explicit bounds on the error incurred with the describing function method. >

Optimal control of input constrained linear systems

Abstract: Explicit, closed-form formulas of optimal and suboptimal control on finite time interval for stochastic, linear, input-constrained systems are derived. The formulas are given in terms of the state transfer matrix of the system, the input constraint, weights of a quadratic criterion and noise variance. For a general system, only implicit solutions can be derived. However, for systems with positive impulse response, an explicit closed-form solution is derived. The optimal control in this case is the control for an unconstrained system with limiting on the input. The limiting function is the random-input describing function of the saturation function. >

Simulation of microwave and millimeter-wave oscillators, present capability and future directions 1

Abstract: The current state-of-the art of oscillator simulation techniques is presented. Candidate approaches for the next genertion of oscillator simulation techniques are reviewed. The method is presented which uses an ecient and robust convolution-based procedure to integrate frequency-domain modeling of a distributed linear network in transient simulation. The impulse response of the entire linear distributed network is obtained and the algorithm presented herein ensures that aliasing eects are minimized by introducing a procedure that ensures that the interconnect network response is both time-limited and band-limited. In particular, articial ltering to bandlimit the response is not required. I. Introduction Large signal simulation of microwave oscillators is necessary to provide steady-state characterization of oscillator performance. Such quantities as power and harmonic content information are then readily available. This is particularly important in achieving rst pass successful design of monolithicly integrated oscillators. Circuit simulation of microwave oscillators by the method of harmonic balance is reasonably mature with several commercial products available and used on a regular basis and been adapted to some rather unusual applications, e.g. [44]. However large signal oscillator analysis in the time domain using programs such as SPICE [20] enables the build-up of oscillations to be observed. In spite of being time-consuming and the diculty of determining the time at which steady state is obtained, time-domain simulation techniques have the ability to predict the start-up of oscillation in addition to the frequency of oscillation and non-steady-state behavior. There a many diculties in applying transient analysis techniques to distributed circuits but these are gradually being addressed. The near future will see a rapid development of these techniques and will be used regularly in microwave oscillator simulation. In this paper we review the current state of microwave oscillator simulation using the harmonic balance approach and describing function methods. We then consider the current state of transient microwave oscillator simulation and focus on transient simulation techniques that have potential for microwave oscillator simulation. We present a new method of transient simulation which integrates the distributed nature of microwave circuits into a transient simulator by using convolution of the impulse response of the linear circuit. Particular attention is given to reducing aliasing eects in deriving the impulse response and in handling of high Q linear circuits.

Transient response analysis of a fuzzy logic controller

Abstract: The exponential-input describing function technique is utilized to investigate the transient response of a fuzzy controller. The symmetric features that fuzzy addition introduces in the decision table of a fuzzy control algorithm made possible the decoupling of the effects of the output error time sequences and, consequently, the representation of the algorithm by multilevel relays. Computer simulations were carried out on a second-order system to assess the performance of the controller, and determine the accuracy of the approximate solution. >

Multiloop controller design for multivariable plants

Abstract: The extension of the SISO (single-input single-output) relay tuning method to a class of multivariable plans is demonstrated. These are diagonally dominant n-by-n plants which are gain stabilizable and with all elements having low-pass characteristics. Gain stabilizability is required in order to lend any purpose to the procedure while the others ensure some accuracy to the information obtained. The requirement of diagonal dominance is made clear. The extension of the theory of describing functions to MIMO (multiple-input multiple-output) systems is given. An example is used to illustrate the analysis. This is followed by preliminary results on controller design which show that Ziegler-Nichols rules can still be used to design multiloop PI (proportional plus integral) controllers for multivariable systems. >

Phase shifter: a nonlinear circuit for direct measurement of phase margin in feedback control systems

Abstract: A nonlinear circuit called phase shifter is introduced. It accurately measures the phase margin of a linear system operating in a feedback control loop. The measurement is made by inserting the phase shifter in the control loop, causing it to undergo self-sustained oscillations (limit cycle) of relatively low amplitude. It is shown that sustained oscillations occur at the gain crossover frequency, and that the phase margin is equal to the phase lag of the inserted circuit. A complete analysis of computing amplitude and frequency of oscillation which employs the classical sinusoidal-input describing function is also given, and the stability of limit cycle and its neighborhood using the theory of integral manifolds is included. >

Approximate Analysis of Nonlinear Systems Driven by Gaussian White Noise

Abstract: In this paper an approximate analysis approach for high order nonlinear systems driven by Gaussian white noise is presented. The algorithm is based on the exact solution to the Fokker-Planck equation of a second order system and as optimal model reduction technique. An illustrative example is given to show the appication of this approximate approach from which it can be seen that the accuracy of the method is better than that of the random describing function method.

Stability Prediction of Nonlinear System Using Multilayer Feed-forward Artificial Neural Network

Abstract: Stability analysis of linear systems can be studied by root-locus technique, Nyquist criterion and Routh Hurwitz's stability testing. Stability study of nonlinear systems can be done by describing function method, phase-plane analysis, numerical integration, and Lyapunov's second method. In this paper, neural network approach to predict the stability of nonlinear systems is presented. This approach is illustrated with an example.

Flutter calculations for a system with interacting nonlinearities

Abstract: Wojciech Potkanski PZL Mielec, Research and Development Center PL-39-300 Mielec, Poland The iterative method is described for flutter analysis concerning nonlinear structures where nonlinearities are present only at finite number of points forming interfaces between linear substructures. The finite amplitude limit-cycle oscillatory motion is to be found. The structural nonlinearities are replaced by so called describing functions carried out by using the harmonic linearization method. The classical flutter equation is employed and only the stiffness matrix is rnodif ied during the iteration process. The sample of numerical results shows the application of the iterative method to the flutter analysis of a glider with nonlinear control system.

Nonlinear control system design based on newly developed dynamic programming algorithm

Abstract: In this paper, a new design method of nonlinear control systems in the time-domain using newly developed iterative dynamic programming algorithm is proposed. This method gives us a straightforward design technique for various nonlinear plants containing nonlinear elements in actuators of plants and/or plants themselves under the forms of optimal control policies of the minimum-time and minimum-fuel-consumption and more general performance criterion functions. This algorithm does not need a trial-and-error procedure. The controller of tabular forms obtained by this method was satisfactorily implemented in various simulations of nonlinear control systems including multiple-input-multiple-output systems. All the controllers designed were tested in the on-line use with the analog simulation of the plants considered. The results obtained here show that this technique is very effective in the design methods of nonlinear control systems. This technique is suggested to overcome difficulties in the use of traditional describing function technique.