Showing papers on "Describing function published in 1988"

On the Frequency Domain Analysis of Nonlinear Systems

Abstract: In this report we study how the Volterra series description of nonlinear systems can be applied to some different problemsWe study oscillations in nonlinear systems It is shown by help of some examples that the Volterra series description gives us the possibility to analyse more general nonlinear systems with respect to oscillations, than is possible by the describing function methodAnother problem we study is how to compensate for nonlinear sensors and sensor noise By means of a simple example we study the question if the sensor linearization or the noise filtering is to be performed firstFinally we discuss by help of examples, the possibility of frequency domain analysis of the nonlinear system that the Volterra series description gives

Development of Nonlinear Building Block Approach

Abstract: A nonlinear building block approach (NLBBA) is proposed to evaluate frequency response characteristics of nonlinear structure systems including springs with nonlinear stiffness and clearances at slide or bearing as occur in actual systems. The advantage of the building block approach (BBA) was that dynamic performance of the total linear system can be evaluated by analyzing and synthesizing the performance of subsystems. In this paper the method was extensively developed to investigate systems with nonlinearities. The describing function was adopted to represent nonlinearity in the system equations. The compliance could be obtained by solving nonlinear simultaneous algebraic equations for multi-degrees-of-freedom system with multinonlinearities. The method was applied to a beam supported by nonlinear springs and a spindle of a machine tool. The evaluated compliance could quantitatively show effects of the nonlinearity such as transfer of the natural frequency, variance of the compliance at the natural frequency, and jump phenomena for sweep of the excitation frequency. The results of the application agreed well with those obtained by step-by-step integration in the time domain (time historical analysis) which is generally used, and also agreed well with the empirical phenomenon of the stability to the self-excited chatter. The computation time could be significantly shortened by the proposed method.

General Criterion And Control Strategy Of Collision-free Movement For Manipulators

Abstract: In this paper, we present a general procedure for finding collision-free paths for a polyhedral object moving through space that is cluttered with obstacle polyhedra. The unified criterion of collision- free motion is represented as Conv {H/sub i/)Conv {O/sub j/)=/spl Oslash/ which is suitable for all cases regardless of the space dimensionality. Based on the criterion, we are led to construct the describing function J which stands for minimal distance between the moving object and obstacles. Tha algorithm derived from describing function J can be used to find any path of interest for both the 2-D problem and the 3-D problem. It is shown that the solution method to the find path problem may be reduced to a search technique. Since the describing function J is related to only some extreme points, the strategy for automatically determining safe paths between configurations in presence of obstacles may be effectively sets implemented by a simple programming.

A New Algorithm for Limit Cycle Analysis of Nonlinear Control Systems

Abstract: A numerical method is presented for the limit cycle analysis of multiloop nonlinear control systems with multiple nonlinearities. Describing functions are used to model the first harmonic gains of the nonlinearities. Existence of a limit cycle is sought by driving the least damped eigenvalues to the imaginary axis. The evolution of the limit cycle is studied next as a function of a critical system-parameter. It is shown that by defining a suitable error function it is possible to use both eigenvalue as well as the eigenvector sensitivities to formulate a generalized Newton-Raphson method to solve simultaneously for the updates of state variable amplitudes in a minimum norm sense. Several case studies have been presented and the development of a numerical procedure to test the stability of the limit cycle has also been reported.

Transient response of joint dominated space structures - A new linearization technique

Abstract: The linearization method presented for calculating the transient responses of nonlinear systems due to initial disturbances is an extension of the 'describing function' approach, in which the system's steady-state response is calculated by representing such nonlinear elements as space structure joints with impedances that are functions of response amplitude. It is shown that, for the transient case, the steady-state impedances can be averaged over the range of responses in order to furnish equivalent values of stiffness and damping; these, for a given set of initial displacements, may be treated as constant during calculations of system response.

On limit cycles in feedback polynomial systems

Abstract: The problem of the existence and of the uncertainty of limit cycles in nonlinear feedback systems is examined. If approximate solution obtained using the describing function method is assumed to be known for a class of multiloop polynomial systems then sufficient conditions are derived to ensure in the neighborhood the existence of a true periodic solution and to evaluate its corresponding bounds. Numerical and graphical techniques are given in order to simplify the application of these results. In particular, for a special class of single-loop systems a procedure called the cone criterion is presented. A number of illustrative examples are provided. >

On an Extension of the Describing Function Method

Abstract: A numerical method is presented for the local search of a limit cycle oscillation in multiloop nonlinear control systems with multiple nonlinearities. The nonlinearities may be asymmetric in nature. Dual inpuit Describing Functions have been used to model the bias and the first harmonic gains of the nonlinearities. Existence of a limit cycle is sought by solving a coupled set of eigenvalue problems. The evolution of the limit cycle is studied next as a function of a critical system-parameter. A Generalized Newton-Raphson method has been developed in this paper to solve for the active updates of the state variable amplitudes in a minimum norm sense. A case study involving a third order control system with an asymmetric nonlinearity is included to illustrate the application of the method.

Simplified realization for generalized transfer functions using modified Markov parameters

Abstract: A simplified algorithm is developed using Taylor series expansion about an arbitrary point, a, for the construction of minimal realizations for generalized transfer functions. The state space realization is given in the form of a linear, time-invariant, discrete-time, singular system. Compared to existing methods, the proposed method reduces the computational effort and memory storage requirements. Furthermore, the quadruple [E, A, B, C] resulting from this technique is more general than similar expressions from existing approaches.

Nonlinear performance of composite-amplifier filters

Abstract: The large-signal characteristics of some composite-amplifier filter structures using two operational amplifiers are analyzed in terms of describing functions Conditions for linearization are developed for both inverting and noninverting configurations Second-order nonlinear effects are minimized and overdriven nonlinear oscillations formulated It is concluded that the price paid for introducing composite amplifiers in active filters is to increase the susceptibility to nonlinear oscillation without improving the dynamic range >

On the limit cycles in feedback bilinear systems

Abstract: The presence of limit cycles in feedback bilinear systems is investigated. Using a procedure quite similar to the sinusoidal describing function method, an approximable solution is derived. The existence and the uncertainty of an actual solution using techniques based on a continuation principle are than stated. >

Evaluation of the large-signal behaviour of active filters

Abstract: A describing function representation of the saturation nonlinearity of operational amplifiers is used to formulate the nonlinear transfer functions of multiple-amplifier filters. The efficacy of various functional approximations for shortening computer time is assessed. Computationally efficient means of predicting jump resonance and nonlinear oscillations are presented, based on the representation. >

Controller design for a robotic manipulator

Abstract: A method for designing a controller for a robotic manipulator is investigated. The objective is to synthesize a controller using only the observable state variables such that the system response follows a given ramp reference input with zero steady state error under a ramp disturbance. The main difficulties are caused by the Coulomb frictions, the spring effect of the link between the drive motor and the manipulator arm, and the unmeasurable state variables. The proposed is based on augmenting the system by additional integrators and feeding back the delayed values of the observable variables. When the closed loop system exhibits a limit cycle, the feedback parameters are readjusted by employing the Describing Function method so that either the limit cycle is completely eliminated or the magnitude of the limit cycle is reduced to an acceptably small level.

Analysis of self-exciting oscillation in nonlinear system with hysteresis characteristics

Abstract: This paper presents a method of numerical analysis by the multiple input describing function for the self-excited oscillation in a nonlinear system with hysteresis. In the traditional analysis by the describing function, the describing function for the hysteresis is determined in an analytic form, and a very complex calculation is required to perform a highly accurate analysis including the harmonic components of the self-excited oscillation. Consequently, in practice, it has been impossible to determine analytically the describing function for the multiple input. Furthermore, the calculation is impossible for the describing function, where the hysteresis characteristics contain a minor loop. From such a viewpoint, this paper proposes a method and algorithm for the analysis of self-excited oscillation. In the proposed method, the describing function is determined numerically by the digital simulation of the hysteresis, and the discrete Fourier transform and its inverse are applied to the result. By this method, the describing function can be determined numerically even for the cases of multiple input and the case with a minor loop. Thus, the self-excited oscillation can be analyzed considering the harmonic components and the multiple mode. As an example, the self-excited oscillation of the nonlinear system is analyzed for the case where the linear part is represented by a second-order transfer function or by a sum of such functions.