Showing papers on "Describing function published in 1986"

Nonlinear dynamical systems

Abstract: State-space models describing function analysis piecewise-linear systems stability control system performance discrete-time systems solutions to exercises.

An Iterative Procedure for Nonlinear Flutter Analysis

Abstract: An iterative procedure in the frequency domain is presented for flutter analysis of large dynamic systems with multiple structural nonlinearities. The major components of the procedure are the describing function approach for system linearization, a structural dynamics modification method for shifting system mode shapes and frequencies, and a complex eigenvalue algorithm for solution of the flutter equation. The purpose of the procedure is to achieve alignment of the oscillatory amplitude in each nonlinear spring with the describing function prediction of stiffness before computing the final stability characteristics. The result is a system tuned to the flutter frequency at the time of instability. To support the development and validation of the procedure, several describing functions are formulated and a quantitative measure of the errors in each is presented. Validation of the iterative method is accomplished through examples involving dynamic systems of increasing complexity, coupled with various representations of the unsteady aerodynamic forces. Both numerical simulations and experimental data are used to compare with the iterative predictions. In the cases studied, the agreement is good to excellent, with the method accurately predicting the amplitude of a limit cycle flutter as well as the initial disturbance required to produce flutter.

A Nonlinear PID Autotuning Algorithm

Abstract: A nonlinear autotuning regulator algorithm is obtained via a direct combination of the Astrom-Hagglund algorithm for the linear case [1] with the sinusoidal-input describing function (SIDEF) approach to nonlinear compensator synthesis of Taylor and Strobel [2]. The basic approach for linear autotuning proceeds as follows: a. install a relay with hysteresis in series with the unknown plant to be controlled; close a unitygain feedback loop around this combination; b. choose several values of hysteresis so that this system exhibits limit cycles; the frequencies and amplitudes of the oscillation at the output of the plant determine points on the plant Nyquist plot; and c. given points on the plant Nyquist plot, set the PID controller gains using an appropriate tuning algorithm (e.g., Ziegler-Nichols). This approach produces good results if the plant is liner or nearly so; however, if the plant behavior is strongly amplitude-dependent, there are likely to be problems with implementing this algorithm. The nonlinear autotuning regulator algorithm which extends the above approach to handle situations where the plant behavior is strongly amplitude-dependent is based on the SIDF approach. In essence, SIDF input/output (I/O) models of the compensated nonlinear system are exploited to directly synthesize a compensator nonlinearity that eliminates or reduces the amplitude dependence of the open-loop I/O relation. The nonlinear synthesis portion of this algorithm is reasonably simple to implement, has been shown to be effective [2], and should be of practical utility. An example application to a precision position control system is provided as an illustration.

Qualitative analysis of oscillations in nonlinear control systems: A describing function approach

Abstract: We present a rigorous analysis of the stability of oscillations in a wide class of nonlinear control systems with numerator dynamics. The analysis employs the classical sinusoidual-input describing function, elementary control theory, and the theory of integral manifolds. We demonstrate, by means of specific examples, how the present results can be used to obtain detailed information concerning the behavior of solutions.

On limit cycles of feedback systems which contain a hysteresis nonlinearity

Abstract: In this paper, we concern ourselves with the stability of limit cycles in feedback systems which have hysteresis nonlinearities. Although the quasi-static analysis of limit cycles (Loeb criterion) predicts, in most cases correctly, the stability properties of limit cycles, it is well known that analyses which are based on the method of describing functions may lead to erroneous conclusions. In this paper, we show to what extent the describing function method can be given a rigorous mathematical basis. We show that for a specific example, the main result of this paper predicts correctly the stability of a limit cycle while the Loeb criterion yields an incorrect result. Also, we show that our analysis explains to a certain extent the presence of distortions in solutions of the class of feedback systems considered herein.In arriving at the main result of this paper, use is made of several known facts for functional differential equations and of a result on integral manifolds.

Some computational techniques for estimating human operator describing functions

Abstract: Computational procedures for improving the reliability of human operator describing functions are described. Special attention is given to the estimation of standard errors associated with mean operator gain and phase shift as computed from an ensemble of experimental trials. This analysis pertains to experiments using sum-of-sines forcing functions. Both open-loop and closed-loop measurement environments are considered.

Analysis of transient oscillations in a nonlinear feedback system

Abstract: In several recent papers [1]-[4] we established rigorous results which indicate when the describing function method is valid (i.e., under which conditions it predicts correctly the existence and the stability of a limit cycle). These results yield also estimates of bounds for limit cycles during transient time periods (i.e., during limit cycle build up or limit cycle build down). In the present paper we address the accuracy of our earlier results. In doing so, we analyzed several typical feedback systems by the method of [1] and we compared these results with those obtained via numerous simulations. Due to space limitations, the results of only one representative feedback system are included.

Computer-aided control engineering environment for nonlinear systems analysis and design

Abstract: We report on recent progress in developing a computer-aided nonlinear control system analysis and design environment based on sinusoidal-input describing function (SIDF) methods. In particular, two major additions have been made to our CAD software for nonlinear controls during 1984: a simulation-based program for generating amplitude-dependent SIDF input/output models for nonlinear plants, and a frequency-domain nonlinear compensator design package. Both of these are described in detail. This software can treat very general nonlinear systems, with no restrictions as to system order, number of nonlinearities, configuration, or nonlinearity type. An overview of the application of this software to the design of controllers for a realistic, nonlinear model of an industrial robot is presented in Taylor (1984). which serves to illustrate the use of these tools. Based on the software presented here, the use of SIDF-based nonlinear control system analysis and design methods is substantially easier to carry out.

Joint nonlinearity effects in the design of a flexible truss structure control system

Abstract: Nonlinear effects are introduced in the dynamics of large space truss structures by the connecting joints which are designed with rather important tolerances to facilitate the assembly of the structures in space. The purpose was to develop means to investigate the nonlinear dynamics of the structures, particularly the limit cycles that might occur when active control is applied to the structures. An analytical method was sought and derived to predict the occurrence of limit cycles and to determine their stability. This method is mainly based on the quasi-linearization of every joint using describing functions. This approach was proven successful when simple dynamical systems were tested. Its applicability to larger systems depends on the amount of computations it requires, and estimates of the computational task tend to indicate that the number of individual sources of nonlinearity should be limited. Alternate analytical approaches, which do not account for every single nonlinearity, or the simulation of a simplified model of the dynamical system should, therefore, be investigated to determine a more effective way to predict limit cycles in large dynamical systems with an important number of distributed nonlinearities.

Linearization Technique for Probabilistic Riser Frequency Domain Analysis

Abstract: This paper presents a rigorous derivation of equations for the minimization of the mean-squared error and defines the describing function for different input components. The technique has a direct application to analyze the dynamic response of marine risers. The describing function method will be used to determine the linearization coefficients for the nonlinear drag effects, taking into account a steady current and the random sea waves. The method will have significant impact on other problems such as fixed offshore platforms and tension leg platforms.

Limit cycles in nonlinear discrete systems

Abstract: The paper gives details of software developed for the investigation of limit cycles in single and multivariable sampled data and digital control systems. Three methods have been implemented, two which use describing function methods and therefore give approximate results and one, which is an extension of Tsypkin's method for relay systems, which gives exact results. Several examples are presented to illustrate the features, relative merits and applications of the methods.

A New Algorithm to Search for a Limit Cycle in Nonlinear Control Systems

Abstract: One normally solves for a limit cycle by solving a set of coupled nonlinear algebraic equations with the aid of a suitable optimization technique. In this paper a new approach is presented to search for a limit cycle locally in nonlinear control systems with multiple nonlinearities. Describing functions are used to derive a quasilinear model for the control system. Eigenvalue and eigenvector sensitivities are then employed to drive the least damped eigenvalue of the quasilinear model to the imaginary axis. It is shown that with the aid of the minimum norm solution one can vary the state variable amplitudes and the system parameters simultaneously. This results in an algorithm with accelerated convergence and improved numerical stability. A simple case study is included and the development of a simple criterion to test the stability of the limit cycle is also reported.

Oscillations of nonlinear feedback systems which contain tightly coupled subsystems in cascade

Abstract: We formulate and prove a theorem which gives a rigorous theoretical justification for the use of describing functions to predict the existence of limit cycles in a multiple nonlinear feedback system and to predict the stability properties of these limit cycles. Our approach uses the classical sinusoidal-input describing function and the theory of integral manifolds. We demonstrate the applicability of our result by means of two specific examples.

A Laboratory Exercise to Illustrate the Describing Function Technique

Abstract: Many control texts do not provide an adequate insight into the assumptions inherent in the describing function technique for analyzing the stability of nonlinear systems. A laboratory experiment is described which provides this background by investigating the harmonic content of the signals in an analog simulation of a third-order system which incorporates an ideal relay. The paper then proceeds to show how the simulation may be extended to investigate the stability of the system when the relay includes a dead zone or hysteresis. It is shown that under these favorable con ditions, the describing function gives good agreement with the results of the simulation.

Suns: the sussex university nonlinear control systems software

Abstract: The paper outlines the capabilities of the Sussex University Nonlinear Software. The programs based on two theoretical techniques, the describing function and Tsypkin's method, evaluate limit cycle solutions in both continuous and discrete feedback systems. The software allows the display of solution waveforms at the input to each nonlinearity and an exact orbital stability criterion, for continuous systems, is used for the limit cycles predicted by the Tsypkin method. Most of the software is for fixed structure canonical feedback systems although one of the DF based programs allows analysis of systems of a free structure. The software may also be used for frequency domain design where gain and phase margins of compensation schemes may be calculated by finding the minimum amount of additional gain or time delay required for a system to limit cycle.

Closed-loop, pilot/vehicle analysis of the approach and landing task

Abstract: In the case of approach and landing, it is universally accepted that the pilot uses more than one vehicle response, or output, to close his control loops. Therefore, to model this task, a multi-loop analysis technique is required. The analysis problem has been in obtaining reasonable analytic estimates of the describing functions representing the pilot's loop compensation. Once these pilot describing functions are obtained, appropriate performance and workload metrics must then be developed for the landing task. The optimal control approach provides a powerful technique for obtaining the necessary describing functions, once the appropriate task objective is defined in terms of a quadratic objective function. An approach is presented through the use of a simple, reasonable objective function and model-based metrics to evaluate loop performance and pilot workload. The results of an analysis of the LAHOS (Landing and Approach of Higher Order Systems) study performed by R.E. Smith is also presented.

An approximation method of describing function for some hysteresis nonlinearities in hydraulic servosystems

Abstract: The describing function method is a more convenient procedure in the investigation of nonlinear feedback control systems. An important condition to use this method is the determination of the describing function of nonlinearity in question. This paper deals with some hysteresis type nonlinearities which are usually encountered in many hydraulic servosystems, and gives a set of suitable approximating formulae in form of very strongly convergent series.