Showing papers on "Describing function published in 1977"

Large amplitude, self-sustained oscillations in difference equations which describe digital filter sections using saturation arithmetic

Abstract: The possibility of oscillations due to adder overflow in digital filter sections of order exceeding two is investigated. Specifically, the following problem is posed: is the property that large amplitude limit cycles cannot be self-sustaining in saturation arithmetic special to second-order sections, or is it also true for some or all higher order sections? This issue is resolved and, beyond that, some new and rather interesting results are presented on the stability implications of approximating a nonlinear element in standard filter structures by a range of linear gains. For every order of the section beyond two we are able to identify a set of coefficients corresponding to an absolutely stable underlying linear system, and a set of initial conditions for which the solution (to the nonlinear recursion) is a limit cycle. As an aid to understanding this behavior, we show that a particular natural class of associated linear recursions is always stable if and only if their order does not exceed two. Finally, motivated by the above result and by the viewpoint of describing functions, we make a conjecture according to which these oscillations do not exist if every member of the above stated class of linear systems is absolutely stable. We show that even this conjecture is false for recursions of order three. Thus, to ensure that overflow oscillations do not occur in high-order sections in which saturation arithmetic is used particular care has to be exercised on a case by case basis.

Spurious predictions of limit cycles in a non-linear feedback system by the describing function method

Abstract: It is known that the describing function method can fail to predict limit cycles which do in fact exist. We wish to draw attention to a different mode of failure, where the method predicts limit cycles which do not exist. A surprising feature of the example we give is that the linear part of the system has excellent low-pass properties, unlike the linear parts used in examples of the other failure mode.

Characterisation of nonlinear 2-ports for the design of class-C amplifiers

Abstract: A method of characterising a nonlinear 2-port based on the `multiple-sinusoid-input describing function? technique is proposed. For simplicity, the effects of the third and higher harmonics are ignored and the nonlinear 2-port is treated as a linear 4-port. The S-parameters of the 4-port are measured, and the `effective? 5-parameters of the nonlinear 2-port are expressed in terms of the 2nd-harmonic terminations. The effects of the 2nd-harmonic reactance terminations on the fundamental-frequency 5-parameters are investigated, and a strategy for optimising the performance of a class-C amplifier is described.

Describing functions, limit cycles and reduced models in nonlinear systems

Abstract: A criterion for the absence of limit cycles in nonlinear multivariable systems is proposed. It is very flexible and can give less conservative results. Comparison is made with existing criteria, and it is also shown that the criterion has great attractions when reduced models are used.

A dual-loop model of the human controller in single-axis tracking tasks

Abstract: A dual loop model of the human controller in single axis compensatory tracking tasks is introduced. This model possesses an inner-loop closure which involves feeding back that portion of the controlled element output rate which is due to control activity. The sensory inputs to the human controller are assumed to be system error and control force. The former is assumed to be sensed via visual, aural, or tactile displays while the latter is assumed to be sensed in kinesthetic fashion. A nonlinear form of the model is briefly discussed. This model is then linearized and parameterized. A set of general adaptive characteristics for the parameterized model is hypothesized. These characteristics describe the manner in which the parameters in the linearized model will vary with such things as display quality. It is demonstrated that the parameterized model can produce controller describing functions which closely approximate those measured in laboratory tracking tasks for a wide variety of controlled elements.

Error bounds for approximate analysis of self-oscillations in autonomous relay control systems†

Abstract: A quantitative technique is presented for determining the existence and uniqueness of simple limit cycles in autonomous relay control systems. In addition, the technique gives bounds on the error resulting from the use of an approximate limit cycle analysis method. The technique here is based on the contraction mapping fixed-point theorem. The intention of this work is to provide an organized procedure for studying error-bound problems in relay control systems in which any computations are as transparent as possible. The results presented here are general in the sense that they may be applied to any approximate analysis, including but not limited to the describing function, in which the relay characteristic may be frequency dependent.

Step Response of Second Order Servo Incorporating Piecewise Linear Element : Case of Hysteresis

Abstract: An exact solution in mathematical sense for the step response of the second order servo incorporating a hysteresis element is obtained. It is made clear that the step responses are sorted out into three types according to the value which the system gain constant takes. And both the condition for the system to have a cycle and an elementary proof that the cycle is a limit cycle are given by the successor function method. Besides the approximate solution for the cycle obtained by the describing function method is studied in detail.

Comments on "Limitations of the describing function method"

Abstract: Recently an example to illustrate the misuse of the describing function method was pointed out It is the purpose of this correspondence to show that the order of the harmonic for which there is an appreciable output is quite high for the same example Further, it is pointed out that the fifth harmonic is not the harmonic with the largest amplitude, but it is the third harmonic Incidentally, a simulation under CSMP on the digital computer showed that the radian frequency of the limit cycle is about ten percent less

Transient Response of Second Order Relay Servomechanism

Abstract: An exact solution in mathematical sense for the transient response of a second order relay servomechanism is obtained. By applying the succession function method, the condition for the system to have a cycle is given and the proof that this cycle is a limit cycle is established. Based on this proof an algorithm and a chart to calculate the period and the amplitude of the cycle are constructed. Moreover, the approximate periodic solution obtained by the describing function method is also studied.