Dual-input Describing Function Research Articles

Characterization of Limit Cycle Oscillations Induced by Fixed Threshold Samplers

In this work, a generalized study of the conditions for the appearance of limit cycle oscillations induced by any kind of sampler with multilevel fixed thresholds is presented. These kinds of samplers, which will be referred to as Fixed Threshold Samplers (FTS), are characterized by a series of parameters, which, when selected properly, allow obtaining some of the most used forms of quantization in Event-Based Control (EBC). Because of some sampler characteristics, the obtained limit cycle oscillations can present a bias, therefore, to characterize them the Dual Input Describing Function (DIDF) method is used. The obtained DIDF is analyzed revealing some interesting properties allowing to simplify the robustness analysis. The analysis takes into account the effect of the disturbance and reference signal influence on the system, generally overlooked in DF analysis. Guidelines about how to perform the robustness analysis are given, showing their application through some study cases.

Designing Limit-Cycle Suppressor Using Dithering and Dual-Input Describing Function Methods

This paper described a method to design a limit-cycle suppressor. The dithering technique was used to eliminate self-sustained oscillations or limit cycles. Otherwise, the Dual Input Describing Function (DIDF) method was applied to design dither parameters and analyze the existence of limit cycles. This method was done in a nonlinear system with relay nonlinearity using three standard dither signals, namely sine, triangle, and square waves. The aim of choosing varying dithers was to investigate the effect of dither shapes and the minimum amplitude required for the quenching strategy. First, the possibility and amplitude of limit cycles were determined graphically on the DIDF curve. Then, the minimum amplitude of dither was calculated based on the DIDF analysis. Finally, a simulation was built to verify the analytical work using a digital computer. The simulation results were related to the analysis results. It was evident that the dithering technique is a simple way to suppress limit cycles in a nonlinear system. This paper also presented that dither is an amplitude function, and square-wave dither has the minimum amplitude to quench limit cycles.

Identification of DC–DC buck converter dynamics using relay feedback method with experimental validation

Accurate dynamics of power converters are necessary to achieve good control performance. In this study, the dynamical model of the DC-DC buck converter is identified by the relay feedback method. The relay is connected in the closed loop to produce a limit cycle output. The important information of the oscillatory output is used for the identification. The relay is approximated using dual-input describing function (DIDF) in the mathematical modelling. DIDF can handle symmetric and asymmetric limit cycle outputs. The converter is modelled as a second-order plus dead-time system. Using the gain and phase angle criteria, analytical expressions are derived to estimate the dynamics. The converter dynamics obtained from the proposed method are compared with that estimated using the state-space averaging method. The model is also identified from the real-time experiment. To check the efficacy of the identified model, a model validation test is performed.

DIDF를 적용한 PID 제어기의 파라미터 설정법 - 불감시간을 가지는 불안정한 시스템의 경우

Though various techniques have been studied as a way of adjusting parameters of PID controllers, no perfect method of determining parameters is available to date. Especially the deign of PID controller for unstable processes with dead time(UPWDT) is even more difficult due to various reasons. Generally the existing design procedures for UPWDT involve deriving formulas to meet gain and phase margin specifications, or using inner loop to stabilize UPWDT before applying PID controller. In this paper, the dual-input describing function(DIDF) method is proposed, by which the performance and robustness of the closed-loop system can be improved. The method is based on moving the critical point (-1+j0) of Nyquist stability to a new position arbitrarily selected on the complex plane. This can be done by determining appropriate coefficients of the DIDF. As a result, we can easily determine parameters of PID-type controller by using existing conventional tuning methods for stable or unstable systems. Simulation results are included to show the effectiveness of the proposed method.

DIDF를 이용한 PID제어기의 성능향상에 관한 연구

Though various techniques have been studied as a way of adjusting parameters of PID controllers, no perfect method of determining parameters is available to date. This paper proposes a new method for enhancing performance of PID controllers by using the characteristics of dual-input describing function (DIDF). In other words, if nonlinear elements with two inputs (DIDF) are connected in series to the plant, the critical point (-1+j0) for Nyquist stability theory can be moved to a position arbitrarily selected on the complex plane by determining necessary coefficients of the DIDF appropriately. This makes the application of the existing conventional PID parameter tuning methods a lot easier, and stability and robustness of the system are improved simultaneously due to the DIDF inserted.

Periodic Disturbance Cancellation by using Dual-Input Describing Function (DIDF) Method

The issue of rejecting periodic disturbances arises in various applications dealing with rotating machinery. A new method using DIDF (Dual-Input Describing Function) is presented for the rejection of periodic disturbances with uncertain frequency. This can be added to an existing feedback control system without altering the closed-loop system stability. The objective is to design a nonlinear compensator to secure specified oscillation amplitude and frequency which are the same as disturbances. We suggest two procedures to determine coefficients for DIDF's synthesis. The structure of the proposed DIDF is so simple that we can easily synthesize. A number of computer simulations were carried out to demonstrate the salient feature of the proposed DIDF compared to the conventional ones(that is, adaptive algorithms).

Stability of single sign integral pulse frequency modulation control systems

Developed herein are detailed mathematical expressions for the stability analysis of single sign integral pulse frequency modulation (SSIPFM) control systems by the dual input describing function (DIDF). The necessary conditions, the d.c. condition and the a.c. condition, for producing limit cycle oscillations in the system, under zero or step inputs, are presented. This work suggests that for such a system it is more efficient to use the DIDF than the DF (describing function) method. The advantages of using this particular method in studying the stability of certain neurophysiological control systems is emphasized.

A Non-Linear Electronic Compensator for Automatic Control Systems

The technique for the synthesis of the dual-input describing function (DIDF) is introduced here and the construction of a non-linear electronic compensator (N-L.E.C.), based on this techniques, is proposed. The above non-linear network is used to compensate . linear and non-linear systems for instability, limit Cycles, speed of response and the static accuracy. Four practical examples are given at the end of this work.

Dual-input describing function of ‘backlash type’ double-valued non-linearities

In the present, paper general relations concerning the Dual Input Describing Functions (DIDF) and the modified characteristics of the ‘ backlash type ’ non-linearities are derived. The calculation of the DIDF of the given characteristic is based on the mathematical expression of the characteristic alone without involving the output waveform

Dual-input describing function of ‘switching-type’ double-valued non-linearities†

Abstract The ‘switching-type’ double-valued non-linearities are defined and general relations concerning their modified characteristics and their dual-input describing functions (DIDF) for two sinusoidal inputs with unrelated frequencies are derived. An illustrative example is also presented.

Steady state analysis of non-linear systems and multiple input describing functions (M.I.D.F.)

The describing function technique is considered as a special case of the more general problem of determining the amplitude of any frequency component in the output of a single valued non-linearity with a multiple frequency input. By using a power series, Fourier transform, and Fourier series, respectively, to represent the non-linear input-output characteristic, three formulas for the amplitude of any output frequency component are derived. The formula based on the Fourier series representation of the non-linearity is presented in detail with a discussion of numerical techniques for very rapid computation of output amplitudes. These methods are generalized for cases when the input frequencies are commensurate. It is shown that the formula for any output amplitude in this case is given by a series which converges rapidly, allowing efficient numerical computation. As an important application, in automatic control theory, of the above methods, both the exact representation and an efficient numerical approximation for the Multiple Input Describing Function (M.I.D.F.) are presented for the case of commensurate as well as incommensurate frequencies. To illustrate these methods, a subharmonic Dual Input Describing Function (D.I.D.F.) is considered, and its relationship to the D.I.D.F. for incommensurate input frequencies is established.

Signal Stabilization of a Memory Type Nonlinearity

The dual input describing function (DIDF) is derived here by statistical methods for the difficult case of a memory type nonlinearity. Special mathematical techniques are used to simplify the problem so that closed form solutions can be obtained. The DIDF is employed to determine the effect of an external sinusoidal signal on system limit cycles. The theoretical results are verified by tests on an analogue computer.

The Response of Nonlinear Closed-Loop Systems to Random Inputs

A dual-input describing function (DIDF) is derived for sine waves and Guassian noise. The derivation follows the correlation method used in [1]. In this paper only single-valued nonlinearities are discussed but extension to multivalued nonlinear elements appears possible. The DIDF is used to investigate the stability and closed-loop response of nonlinear systems excited by random noise. Previous investigations have provided only for the random component at the input to the nonlinear element. It is shown that previous work is invalid insofar as it neglects the possibility of oscillations in the nonautonomous system if the autonomous system is stable and vice versa. Two examples are presented which show (i) the necessity of the DIDF approach for systems which are stable without input, and (ii) the possibility of successfully obtaining stable response to certain classes of inputs with systems which appear unstable without inputs. The present investigation is an extension of the authors’ previous work on the stability and closed-loop response of nonlinear systems excited by sinusoidal inputs [2].

A New Dual-Input Describing Function and an Application to the Stability of Forced Nonlinear Systems

A new general DIDF (dual input describing function) has been analytically derived for single-valued nonlinearities subjected to two arbitrary noncommensurate sine waves. The development corroborates a previous approximate development for two sine waves widely separated in frequency. The new DIDF is applied to the problem of the stability of nonlinear systems subjected to sinusoidal forcing. It is shown that the conventional DF (describing function) cannot be used for nonautonomous systems without additional safeguards. After it has been shown by the DIDF that no auto-oscillations exist for given input conditions it is proper to employ the conventional DF in any of the methods suggested in the literature to obtain the closed-loop frequency response under those conditions.

Frequency response of feedback relay amplifiers

The frequency response of feedback relay amplifiers is analysed by the dual-input describing-function method. At low signal frequencies, the amplifier behaves like a saturating linear amplifier. This picture is modified at higher frequencies, owing to synchronous effects, and it is difficult to achieve satisfactory operation when the ratio of the input-signal frequency to the periodic excitation frequency is appreciable. To eliminate synchronous effects within the signal band, restrictions must be imposed on the forward linear-transfer function. The theory is compared with an experimental example.

Discussion on “Extension of the dual-input describing-function technique to systems containing reactive non-linearity”

Discussion on “Extension of the dual-input describing-function technique to systems containing reactive non-linearity”

Extension of the dual-input describing-function technique to systems containing reactive non-linearity

Kochenburger's describing function and also the dual-input describing function due to West, Douce and Livesley are well known. These techniques enable the graphical solution of nonlinear differential equations in which the non-linear coefficient is associated with the first-order differential term to be obtained. The paper describes an extension of these methods, allowing the non-linear coefficient to be associated with any term in the differential equation.It is shown that the typical non-linearity presented by an iron-cored inductor or by a ferro-electric capacitor may be resolved into a simple feedback system containing a single non-linear element which is independent of frequency, together with elements possessing linear transfer functions. The resulting equivalent block diagram is suitable for analysis by either of the describing-function techniques. As an example, the dual-input technique is used to predict accurately jump phenomena in an iron-cored ferroresonant circuit, and the validity of the equivalent block diagram suggested is also checked by simulation on an analogue computer.