Sinusoidal-input Describing Function Research Articles

A Nonlinear Integrator Based on the First-Order Reset Element

This work proposes a novel nonlinear Proportional-Integral (PI) controller, which utilizes a generalized first-order reset element. The proposed element can achieve similar magnitude-characteristics as its linear counterpart but with less phase lag at the open-loop crossover frequency (i.e. the control bandwidth), according to a sinusoidal-input describing function (SIDF) analysis. The same can be achieved with an existing reset-based integrator, the Clegg integrator (CI). However, it is known that a Proportional-CI (PCI) element can excessively generate higher-order harmonics of its input, which are neglected in the SIDF-analysis. Furthermore, a PCI can cause a limit cycle when placed in closed-loop with certain types of plants. The novel PI controller proposed in this work can prevent the limit cycle and can reduce the generation of higher-order harmonics, while retaining the beneficial phase advantage that is associated with existing reset-based PI controllers.

Impact of Governor Non-Linearities Representation on Power System Ultra-Low Frequency Oscillations

Power system dynamic models are essential for the stability analysis of a perturbed electrical system. In recent years, hydro-dominant power systems have experienced the occurrence of ultra-low frequency oscillations (ULFO). While power system dynamic simulation stands as an essential tool for the research community to analyze and understand this phenomenon, such systems involve non-linearities in their dynamic model affecting the ULFO analysis’s validity. This present work investigates the impact of governor non-linearities representation on the validity of the ULFO analysis. Sinusoidal-input describing function (SIDF) is borrowed from control theory to appreciate existing governor non-linearities models’ capacity to respond accurately to ULFOs. Through rigorous test-cases, results prove the imperativeness of prior accurate measurement of those governor non-linearities before their numerical implementation to ensure the power system dynamic simulation model’s validity for ULFO analysis.

Numerical Solution to the Inverse Sinusoidal Input Describing Function

A computational solution to the inverse sinusoidal input describing function (SIDF) for a broad class of static nonlinearities is presented. The proposed numerical solution uses the SIDF gain and phase distortions to identify the nonlinearity. This solution, unlike existing methods in the literature, does not require a priori knowledge of the nonlinearity structure in the estimation process and is applicable to both single- and double-valued nonlinearities. The output from the algorithm is a nonparametric model of the nonlinearity from which a parametric model can be recovered by least-square estimation (LSE) method. Three examples are presented to validate the proposed algorithm.

Graphical user interface for input output characterization of single variable and multivariable highly nonlinear systems

This paper presents a Graphical User Interface (GUI) software utility for the input/output characterization of single variable and multivariable nonlinear systems by obtaining the sinusoidal input describing function (SIDF) of the plant. The software utility is developed on MATLAB R2011a environment. The developed GUI holds no restriction on the nonlinearity type, arrangement and system order; provided that output(s) of the system is obtainable either though simulation or experiments. An insight to the GUI and its features are presented in this paper and example problems from both single variable and multivariable cases are demonstrated. The formulation of input/output behavior of the system is discussed and the nucleus of the MATLAB command underlying the user interface has been outlined. Some of the industries that would benefit from this software utility includes but not limited to aerospace, defense technology, robotics and automotive.

VIBRATION INDUCED FAILURE ANALYSIS OF A HIGH SPEED ROTOR SUPPORTED BY ACTIVE MAGNETIC BEARINGS

Active Magnetic Bearings (AMBs) are increasingly used in various industries and a quick re-levitation of AMBs supported high speed flexible rotor is necessary in case of vibration induced failure. A robust fault diagnosis algorithm is presented to detect suspected saturation type of nonlinearity associated with a power amplifier. A five degree-of-freedom AMB system consisting of four opposing pair of radial magnets and a pair of axial magnets is considered. In this paper failure of an industrial grade AMB system is investigated using Sinusoidal Input Describing Function (SIDF) method. SIDF predicts the gain and frequency at which failure occurs. It is demonstrated that the predicted frequency is in agreement with the frequency at which failure occurs.

New Methodology for Control of Nonlinear Unstable Multivariable Systems

AbstractIn this paper, a new methodology for control of nonlinear unstable multivariable systems is developed. The method involves stabilization of the nonlinear plant followed by generation of sinusoidal input describing function (SIDF) models of the stabilized closed-loop system. With the known SIDF model of the closed-loop system and the known stabilizing controller, the model of the unstable nonlinear multivariable system is extracted; then a controller for the model is designed. Finally, the design is verified by simulation. Stability is demonstrated by successful generation of SIDF models of the designed closed-loop system for various amplitudes of excitation. The method is applied to a two-input/two-output (TITO) model of a nonlinear unstable robotic arm, and the results are compared with an alternative approach.

Optical Parametric Oscillator Longitudinal Modes Suppression Based on Smith Predictor Control Scheme

Fiber optical parametric oscillators (FOPOs) contain multiple longitudinal modes arising from the hundreds of meters to a kilometer scale length of their cavity. To obtain better understanding of the longitudinal modes and to find out how to suppress them, a model for the operation of FOPOs is introduced. This model utilizes the sinusoidal-input describing function (SIDF) as a quasi-linear approximation of the nonlinear dynamics of parametric amplification. Applying this model, the SIDF of the FOPO is derived, which predicts that the origin of the longitudinal modes is the presence of the cavity round-trip delay term in the characteristic equation (the denominator) of the transfer function. To eliminate the effect of the delay term, a modified Smith predictor control scheme has been applied to the FOPO cavity, and a control algorithm is developed. This algorithm has been numerically analyzed and experimentally implemented, where the results indicate drastic suppression of the longitudinal modes.

Nonlinear Lead-Lag Controller Synthesis for Unstable Nonlinear Systems

AbstractA new systematic procedure for synthesis of nonlinear lead-lag controllers for highly nonlinear unstable plants is developed. The procedure is to stabilize the unstable plant followed by generation of sinusoidal-input describing function (SIDF) models at various operating regimes of interest. The unstable plant frequency domain behavior is extracted, and a set of linear lead-lag controllers are designed that would force the open-loop behavior of the system (comprised of the lead-lag controller and the nonlinear plant) and mimic that of desired. Finally, a describing function inversion is used, and the nonlinear functions describing the values of the lead-lag coefficients as a function of the error signal are obtained. Experimental results are presented for verification, and bounded-input bounded-output stability is demonstrated by successful generation of the describing function models of the closed-loop system comprised of the synthesized nonlinear lead-lag controller and the nonlinear plant.

Limit Cycle Prediction Based on Evolutionary Multiobjective Formulation

This paper is concerned with an evolutionary search for limit cycle operation in a class of nonlinear systems. In the first part, single input single output (SISO) systems are investigated and sinusoidal input describing function (SIDF) is extended to those cases where the key assumption in its derivation is violated. Describing function matrix (DMF) is employed to take into account the effects of higher harmonic signals and enhance the accuracy of predicting limit cycle operation. In the second part, SIDF is extended to the class of nonlinear multiinput multioutput (MIMO) systems containing separable nonlinear elements of any general form. In both cases linearized harmonic balance equations are derived and the search for a limit cycle is formulated as a multiobjective problem. Multiobjective genetic algorithm (MOGA) is utilized to search the space of parameters of theoretically possible limit cycle operations. Case studies are presented to demonstrate the effectiveness of the proposed approach.

EVOLUTIONARY SEARCH FOR LIMIT CYCLE AND CONTROLLER DESIGN IN MULTIVARIABLE NONLINEAR SYSTEMS

ABSTRACTA feature of many practical control systems is a Multi‐Input Multi‐Output (MIMO) interactive structure with one or more gross nonlinearities. A primary controller design task in such circumstances is to predict and ensure the avoidance of limit cycling conditions followed by achieving other design objectives. This paper outlines how such a system may be investigated using the Sinusoidal Input Describing Function (SIDF) philosophy quantifying magnitude, frequency and phase of any possible limit cycle operation. While Sinusoidal Input Describing function is a suitable linearization technique in the frequency domain for assessment of stability and limit cycle operation, it can not be employed in time domain. In order to be able to incorporate the time domain requirements in an overall controller design technique, the appropriate linearization technique suggested here is the Exponential Input Describing Function (EIDF).First, an evolutionary search based on a multi‐objective formulation is employed for the direct solution of the harmonic balance system matrix equation. The search is based on Multi‐Objective Genetic Algorithms (MOGA) and is capable of predicting specified modes of theoretically possible limit cycle operation.Second, the design requirements in time as well as frequency domain are formulated by a set of constraint inequalities. A numerical synthesis procedure also based on Multi‐Objective Genetic Algorithm is employed to adjust the initial compensator parameters to meet the imposed constraints. Robust stability and robust performance are investigated with respect to linearization uncertainty within the context of multiobjective formulation. In order to make the Genetic Algorithm (GA) search more amenable to design trade‐off between different and often contradictory specifications, a weighted sum of the functions is introduced. This criterion is subsequently optimized subject to the nonlinear system dynamics and a set of design requirements. Examples of use are given to illustrate the effectiveness of the proposed approach.

Convergence and computation of describing functions using a recursive Volterra series

AbstractPresented in this paper is a comparison of algorithms for computing an approximation to the sinusoidal input describing function (SIDF) for the nonlinear differential equation ẏ(t)+b1y(t)+b2u2(t)y(t) = K(u̇(t)+b3u(t)) The importance of this nonlinear differential equation comes from the context of nonlinear feedback controller design. Specifically, this equation is either a linear lead or lag controller (depending on the coefficient values) augmented with a nonlinear, polynomial type term. Consequently, obtaining a SIDF representation of this nonlinear differential equation or creating a process to obtain SIDFs for other similar differential equations, will facilitate nonlinear controller design using classical loop shaping tools. The two SIDF approximations studied include the well‐established harmonic balance method and a Volterra series based algorithm. In applying the Volterra series, several theoretical issues were addressed including the development of a recursive solution that calculates high order Volterra transfer functions, and the guarantee of convergence to an arbitrary accuracy. Throughout the paper, case studies are presented. Copyright © 2004 John Wiley & Sons, Ltd.

Input/output characterization of highly nonlinear multivariable systems

This paper presents a software in the MATLAB environment for input/output characterization of highly nonlinear multivariable systems. The software is based on obtaining the sinusoidal-input describing function (SIDF) models of the nonlinear multivariable plant. The mathematical development of the input/output characterization approach is outlined, and the printout of the nucleus of the characterization software is given. The method and the associated software have no restriction on nonlinearity type, configuration, and arrangement, system order as well as the number of inputs and outputs; the only requirement is that system output(s) for sinusoidal input(s) must be obtainable or available by either simulation or experiments. The software has a variety of applications in such industries as robotics, aerospace, automotive, and chemical processes. The method and the associated software are demonstrated by solving a problem of the sort encountered in aerospace.

Robust control of positioning systems with a multi-step bang-bang actuator

A nonlinear control scheme for preventing the limit cycle due to the nonlinearity of the multi-step bang-bang actuator in mechanical position control systems is proposed. A linearized model, sinusoidal input describing function (SIDF), for a multi-step bang-bang actuator is introduced to compensate the nonlinearity of the multi-step bang-bang actuator. Using that model, an H ∞ robust controller for position control systems with a bang-bang actuator is proposed by loop shaping techniques with normalized coprime factorization stabilization to address the robustness. The proposed scheme requires a smaller deadband as a result of compensating the nonlinearity of the bang-bang actuator. A single-axis servo system is implemented in order to verify the proposed control scheme experimentally. Experimental results show that the controller can satisfy the special interest, silent contact switching of the actuator.

H ∞ control analysis and design for nonlinear systems

The μ analysis method is applied to a class of multivariable separable autonomous nonlinear feedback systems. The nonlinear element is replaced by its sinusoidal input describing function (SIDF) to develop a quasilinear nominal model for control analysis and design. Some properties of the ∞ -norm and structured singular values (SSVs) are shown to hold for the type of systems where the output is a function of control input-signal amplitude and frequency. The model uncertainty induced by the SDIF approximation is calculated and incorporated in the control design as an input multiplicative perturbation to the quasilinear nominal model. The μ analysis technique is then employed to study the stability and performance robustness of the nonlinear control system. Using small gain theorem, the condition for the absence of limit cycle is derived. The information obtained from this analysis is then used to design a bank of controllers using the H ∞ optimization-based μ synthesis technique. These controllers are designed such that the closed loop system is robust over a range of control-input amplitude. A simple scheme is then proposed to select the appropriate controller for the operating range. The simulation results illustrate that the proposed technique may be used to design a high performance robust controller for nonlinear systems and the absence of the limit cycle oscillation may be ensured.

CAE Tools for Nonlinear Systems Analysis and Design Based on Sinusoidal-Input Describing Functions

Abstract We report on recent progress in the development of a computer-aided engineering (CAE) environment for nonlinear control system analysis and design based on sinusoidal-input describing function (SIDF) methods. Several major additions have been made to our nonlinear controls CAE software: ACSL macros were developed to allow the generation of SIDF models of nonlinear plants in a manner analogous to that of the SIMNON-based software developed earlier, and MATLAB routines were developed for the analysis of these models and for the design of general nonlinear controllers based on them. This software provides an integrated tool set for treating very general nonlinear systems with no restrictions on system order, number of nonlinearities, configuration, or nonlinearity type. Based on the new software presented here, the use of SIDF-based nonlinear control system analysis and design methods is substantially easier to carry out and more powerful than before

ロボットハンドトルクサーボ系の入力依存形安定性

This paper discusses input-dependent stability on torque servo system of robot hands. With an insufficient space to mount actuators inside the finger joints, a finger joint is normally actuated by a tendon-sheath driving system. Although this driving system enables us to construct a finger system with simple mechanism and large flexibility, friction and compliance existing in tendon-sheath driving system generally bring a nonlinear characteristic involving a hysteresis into the dependence of joint torque output from actuator displacement. While this transmission characteristic is close to the well-known backlash behaviour appearing for the gears system, the author newly found input-dependent characteristics existing in the backlash region of the transmission system through the precise experiments. Namely, the characteristic is that the tendon-sheath transmission system behaves as if there were virtual springs in the backlash region and the strength of virtual spring depending on the input. Furthermore, it was confirmed that there are close relationships between the input-dependent backlash characteristics and the input-dependent stability. Based on the experiments, the transmission characteristics are modelled in simple equations, and the system stability is explored by using the techniques of sinusoidal-input-describing-function (SIDF) . Resultantly, we obtained non-dimensional stability-criterion-maps explaining the experimental results successfully.

Frequency-domain Modeling of Nonlinear Multivariable Systems

A frequency-domain technique for input/output (I/O) characterization of stable, multivariable, and highly nonlinear systems (e.g., industrial robots, aerospace vehicles, chemical processes) is presented. We require only that the nonlinear system is representable in state-variable differential equation form, and that it is possible to integrate the system of equations numerically when input signals are sinusoidal. Otherwise, the technique is not restricted with respect to system order, number of nonlinearities, configuration, or nonlinearity type. The I/O characterization technique involves determining the gain and phase of the nonlinear system response to sinusoidal inputs of various excitation amplitudes at a set of user-defined discrete frequencies. These sinusoidal-input describing function (SIDF) models are obtained by exciting all input channels at one time with sinusoids of different but nearly equal frequencies, integrating the dynamic equations of motion over time, and simultaneously performing a Fourier analysis (evaluating Fourier integrals) on the output signals after they are at steady-state. Repeating this procedure for various amplitude-levels of the excitation signal will result in a number of matrix sinusoidal-input describing function I/O models.

Input/output characterization of highly-nonlinear systems

This paper presents a computer-aided engineering (CAE) environment for input/output characterization of highly nonlinear systems. The approach is based on obtaining the sinusoidal-input describing function (SIDF) models of the nonlinear plant. These models are obtained by exciting the plant with a sinusoid of known amplitude and frequency, integrating the dynamic equations of motion to obtain system output, and finally evaluating Fourier integrals when system output is at steady-state. A software utility based on this technique is developed and outlined. The software is written in standard FORmula TRANslation (FORTRAN) 77, and it is developed on a Harris-800 super-minicomputer and a Tektronix 4115B high resolution, raster, and color graphics terminal. At present, the characterization approach and the associated software is limited to single-input single-output, nonlinear, time-invariant, deterministic, and stable systems.

Computer-Aided Control Engineering Environment for Nonlinear Systems Analysis and Design

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.