Showing papers on "Describing function published in 2017"

Chattering Analysis of HOSM Controlled Systems: Frequency Domain Approach

Abstract: In this article, a methodology of chattering analysis of Sliding Mode/Higher Order Sliding Mode (SM/HOSM) control systems in the frequency domain is presented. A numerical method for computing the Describing Functions (DFs) of HOSM control algorithms is given. The algorithm for predicted chattering parameters in dynamically perturbed system via the Describing Function-Harmonic Balance (DF-HB) technique is proposed. The stability conditions for limit cycles in dynamically perturbed HOSM control systems are presented. The concepts of Practical Stability Phase Margin ( PSPM ) and Practical Stability Gain Margin ( PSGM ) as the robustness metrics to unmodeled dynamics in HOSM control systems are introduced. The methodologies for computing PSPM and PSGM via DF-HB technique are provided. The accuracy of the proposed chattering analysis techniques is confirmed via computer simulations.

Analysis and Suppression of Limit Cycle Oscillation for Transmission System With Backlash Nonlinearity

Abstract: This paper mainly deals with the prediction and suppression of limit cycle oscillation existed in the two-mass system with nonlinear backlash. To improve position control precision of the drive system, the full-closed-loop feedback scheme is considered. Then, a comprehensive and definite prediction of limit cycle is given via the describing function method. In order to compensate backlash nonlinearity, the state feedback control method is employed. To overcome the insufficient application of the state feedback control in the complex nonlinear system, a novel design concept is proposed. The influence of backlash on the nonlinear system is substituted with an additional compensation component to the control signal, making the system approach a linear one. Following this, a pole placement scheme is designed, which aims to establish a feedback structure that can make the system equivalent to a rigid system. In such a case, the limit cycle oscillation can be eliminated; thus, the position control precision can also be enhanced. In all cases, the validity of the proposal is verified by experimental results.

Stability Analysis of the Shunt Regulator With Nonlinear Controller in PCU Based on Describing Function Method

Abstract: The shunt regulator is a key part of the power conditioning unit, which is becoming more and more important in the spacecrafts. Due to the nonlinear link, the switch operating with hysteresis control, the shunt regulator is a nonlinear system; however, the stability analysis method for the system is still grounded on the approximate linearization method, which neglects some of the ac and dc characteristics. In this paper, a novel stability analysis method for the nonlinear systems is proposed based on the describing function, which can be used to model the nonlinear link in the shunt regulator system. Further, the different conclusion of the shunt regulator system stability are drawn, by the approximate linearization method and the stability analysis method proposed in this paper, respectively. Finally, the simulation and experimental results prove the correctness of the stability analysis method with describing function and discover the incompletion of the approximate linearization methods on the stability analysis. Therefore, this paper provides a new way to analyze the stability of the nonlinear systems.

A unified event-based control approach for FOPTD and IPTD processes based on the filtered Smith predictor

Abstract: A new unified design of an event-based PID control architecture for self-regulating and integral processes is investigated in this work. The design is based on the symmetrical-send-on-delta (SSOD) sampling technique and on the filtered Smith predictor (FSP). In particular, the conditions to achieve robust stability to limit cycles are studied from the point of view of the filter parameters through the describing functions theory. In this context, a simple frequency domain-based tuning methodology to address the cases of first-order-plus-dead-time and integrator-plus-dead-time processes is proposed. With this method, the properties of the filter parameters are also explored to cope with the set-point tracking and the load disturbance rejection tasks. The effectiveness of the general approach is evaluated by simulations and experimental tests.

Augmented nonlinear differentiator design and application to nonlinear uncertain systems.

Abstract: In this paper, an augmented nonlinear differentiator (AND) based on sigmoid function is developed to calculate the noise-less time derivative under noisy measurement condition. The essential philosophy of proposed AND in achieving high attenuation of noise effect is established by expanding the signal dynamics with extra state variable representing the integrated noisy measurement, then with the integral of measurement as input, the augmented differentiator is formulated to improve the estimation quality. The prominent advantages of the present differentiation technique are: (i) better noise suppression ability can be achieved without appreciable delay; (ii) the improved methodology can be readily extended to construct augmented high-order differentiator to obtain multiple derivatives. In addition, the convergence property and robustness performance against noises are investigated via singular perturbation theory and describing function method, respectively. Also, comparison with several classical differentiators is given to illustrate the superiority of AND in noise suppression. Finally, the robust control problems of nonlinear uncertain systems, including a numerical example and a mass spring system, are addressed to demonstrate the effectiveness of AND in precisely estimating the disturbance and providing the unavailable differential estimate to implement output feedback based controller.

Consensus of second order multi-agents with actuator saturation and asynchronous time-delays

Abstract: This article presents the consensus of a saturated second order multi-agent system with non-switching dynamics that can be represented by a directed graph. The system is affected by data processing (input delay) and communication time-delays that are assumed to be asynchronous. The agents have saturation nonlinearities, each of them is approximated into separate linear and nonlinear elements. Nonlinear elements are represented by describing functions. Describing functions and stability of linear elements are used to estimate the existence of limit cycles in the system with multiple control laws. Stability analysis of the linear element is performed using Lyapunov-Krasovskii functions and frequency domain analysis. A comparison of pros and cons of both the analyses with respect to time-delay ranges, applicability and computation complexity is presented. Simulation and corresponding hardware implementation results are demonstrated to support theoretical results.

Limit Cycle Prediction of Systems with Fractional Controllers and Backlash

Abstract: This article investigates the limit cycle (LC) prediction of systems with backlash by means of the describing function (DF) when using discrete fractional-order (FO) algorithms. The DF is an approximate method that gives good estimates of LCs. The implementation of FO controllers requires the use of rational approximations, but such realizations produce distinct dynamic types of behavior. This study analyzes the accuracy in the prediction of LCs, namely their amplitude and frequency, when using several different algorithms. To illustrate this problem we use FO-PID algorithms in the control of systems with backlash.

Small signal modeling of dual-edge PWM modulator with fixed clock frequency

Abstract: In this paper, the describing function approach is used to model a dual-edge PWM modulator with a fixed clock frequency. A detailed PWM modulator circuit configuration is introduced with the derivation of the model. Good matching is observed when comparing the analytical model and SIMPLIS simulation. System level analysis demonstrates the advantages of the proposed model versus a conventional average model. The proposed model enables a more accurate compensator design for predicting the dynamic characteristics of the entire system.

Modeling and analysis of phase-shift controlled LCL resonant converter in wireless charging systems

Abstract: In this paper, a simplified low order linear model for LCL compensated WCS is presented using Extended Describing Function (EDF) technique. It turns out that the six-order LCL type WCS behaves as a second-order under-damped system within low frequency range. The simplified reduced order model provides intuitive understanding into both steady and dynamic performance of the converter and also facilitates controller design. The stability boundary and ripple suppression condition using PI controller is given. Considering the limitation of conventional PI controller, the improved PI combining notch filter is proposed to acquire minimal charging ripple and quick dynamic response simultaneously. Experimental results are presented to verify the accuracy and robustness of proposed model and control design.

Consensus of second order multi-agents with actuator saturation and asynchronous time-delays

Abstract: This study presents the consensus of a saturated second-order multi-agent system with non-switching dynamics that can be represented by a directed graph. The system is affected by data processing (input delay) and communication time-delays that are assumed to be asynchronous. The agents have saturation non-linearities, each of them is approximated into separate linear and non-linear elements. Non-linear elements are represented by describing functions. Describing functions and stability of linear elements are used to estimate the existence of limit cycles in the system with multiple control laws. Stability analysis of the linear element is performed using Lyapunov-Krasovskii functions and frequency domain analysis. A comparison of pros and cons of both the analyses with respect to time-delay ranges, applicability and computation complexity is presented. Simulation and corresponding hardware implementation results are demonstrated to support theoretical results.

Limit cycle oscillation and multiple entrainment phenomena in a duffing oscillator under time-delayed displacement feedback:

Abstract: The Duffing oscillator under time-delayed displacement feedback is investigated to study the effect of intentional time-delay on the global dynamics of the oscillator. From the free vibration study performed by employing the describing function method it is observed that for the undamped oscillator, an infinite number of limit cycles is present for all possible values of gain and delay. The number of stable and unstable limit cycles in the gain versus delay plane is studied region wise with the help of limit cycle stability lines. Secondly, in a damped system, the number of limit cycles is finite and depends upon the values of gain, delay and damping coefficient from which the maximum number of limit cycles, their frequencies and amplitudes are obtained. When the system is excited by harmonic forcing, these limit cycles exhibit the phenomena of multiple entrainments and their frequency response curves become very complex and most often results in the very high amplitude oscillations. The study of the forc...

Analyzing Oscillators using Describing Functions

Abstract: In this manuscript, we discuss the use of describing functions as a systematic approach to the analysis and design of oscillators. Describing functions are traditionally used to study the stability of nonlinear control systems, and have been adapted for analyzing LC oscillators. We show that they can be applied to other categories of oscillators too, including relaxation and ring oscillators. With the help of several examples of oscillators from various physical domains, we illustrate the techniques involved, and also demonstrate the effectiveness and limitations of describing functions for oscillator analysis.

Nonlinear control and stability analysis of a stroke limited inertial actuator in velocity feedback

Abstract: Inertial actuators are active devices used, with velocity feedback controllers, to reduce structural vibrations. Physical limits, such as stroke saturation, can affect the behaviour and the stability of the control system. In particular, limit cycle oscillations are observed. In this paper, we propose a nonlinear control strategy to prevent the destabilisation of the velocity feedback loop due to stroke saturation. A time domain model of the stroke limited inertial actuator mounted on a single degree of freedom structure is derived. The stability of the nonlinear system under a velocity feedback control with fixed gain is investigated using the describing function method for the detection of limit cycles. The outcomes are also verified by time domain simulations. The presented nonlinear controller increases the stability region of the system compared to only the velocity feedback controller.

An Improved Describing Function With Applications for OTA-Based Circuits

Abstract: Electronic systems make extensive use of operational transconductance amplifiers (OTAs) to build filters and oscillators. Studying the effects of the saturation nonlinearity on these OTA-based circuits is difficult and often requires lengthy simulations to check the system’s performance under large-signal operation. The describing function (DF) theory allows to circumvent these simulations by deriving a signal-dependent linearized gain, which predicts the effects of the nonlinearity. However, its use is limited since the state-of-the-art DFs deviate significantly from the real saturating behavior of OTAs. This paper proposes an improved DF, which can be directly derived from the static nonlinear characteristic of the transconductance amplifier. The performance of the proposed methodology is demonstrated for both an OTA-based filter and oscillator. It is shown that the proposed DF has a better nonlinear prediction capability than the state-of-the-art solutions.

Iterative solution of the dynamic responses of locally nonlinear structures with drift

Abstract: Considering the operating point drift in the dynamic responses of locally nonlinear structures, a new iterative method based on describing functions is introduced to solve for the steady-state frequency response. Drift in the operating point arises in the presence of asymmetric nonlinearity, pre-deformation or static loads. In this study, the internal nonlinear forces are expressed using describing functions. The complex equations governing the responses of multiple frequency components are established and are iteratively solved using the Inverse Matrix Update Method. The nonlinear frequency responses can thus be rapidly obtained, and the validity of the method is verified by simulations. The presented method can be applied to large-scale structures with multiple nonlinear elements.

Limit Cycle Oscillation Amplitude Tailorng Based on Describing Functions and $$\mu $$ Analysis

Abstract: Freeplay is a nonlinearity commonly encountered in aeroservoelastic applications which is known to cause Limit Cycle Oscillations (LCOs), limited amplitude flutter phenomena not captured by a linear analysis. Uncertainties in the models are also known to play an important role in triggering instabilities which might not be present in the nominal case, or altering their features in an unpredictable way. This paper shows the process to build a framework to study the nonlinear behavior of a typical section affected by a freeplay in the control surface and uncertainties in its parameters’ values. Starting from the definition of the nominal aeroelastic model, the nonlinear framework is implemented by means of the Describing Function (DF) method and robust analysis is introduced by means of \(\mu \) technique. In addition, it is shown an idea to perform a tailoring of the LCO graph of the system with the practical goal to limit the oscillation amplitude. Implications and advantages of using DF and \(\mu \) as primary tools are highlighted, and prowess of the methodology is showcased with an example.

Time Domain Stability Margin Assessment of the NASA Space Launch System GN&C Design for Exploration Mission One

Abstract: The baseline stability margins for NASA's Space Launch System (SLS) launch vehicle were generated via the classical approach of linearizing the system equations of motion and determining the gain and phase margins from the resulting frequency domain model. To improve the fidelity of the classical methods, the linear frequency domain approach can be extended by replacing static, memoryless nonlinearities with describing functions. This technique, however, does not address the time varying nature of the dynamics of a launch vehicle in flight. An alternative technique for the evaluation of the stability of the nonlinear launch vehicle dynamics along its trajectory is to incrementally adjust the gain and/or time delay in the time domain simulation until the system exhibits unstable behavior. This technique has the added benefit of providing a direct comparison between the time domain and frequency domain tools in support of simulation validation.

Analysis of orbital stability in lure system, based on dynamic harmonic balance

Abstract: A criterion of orbital stability of a limit cycle in a Lure system is formulated using the dynamic harmonic balance (DHB) principle and the describing function (DF) method. It is demonstrated that a more precise formulation of orbital stability than that provided by the Loeb's criterion can be produced. This enhancement is achieved through elimination of the assumption that is used in Loeb's criterion derivation. It is demonstrated in the paper that this assumption does not hold. An example of analysis is given.

Adaptive control of tonal disturbance in mechanical systems with nonlinear damping

Abstract: This paper describes an adaptive system for controlling the tonal vibration of a single-degree-of-freedom system with nonlinear damping. The adaptive control system consists of a force actuator in parallel with the suspension, which includes the nonlinear damper, and a velocity sensor mounted on the mass. The adaptation of the controller is done once every period of the excitation. Because the response of the nonlinear system changes with excitation level, conventional adaptive algorithms, with a linear model of the plant, can be slow to converge and may not achieve the desired performance. An on-line observer is used to obtain a describing function model of the plant, which can vary with the excitation level. This allows the adaptive control algorithm to converge more quickly than using a fixed plant model, although care has to be taken to ensure that the dynamics of the observer do not interfere with the dynamics of the adaptive controller.

Robustness of adaptive control systems to unmodeled dynamics : a describing function viewpoint

Abstract: Recently, it was shown that a standard model reference adaptive controller modified with a suitably tuned projection algorithm ensures robustness (global boundedness) of the overall adaptive system to a class of unmodeled dynamics with minimal restrictions. However, extensive first-principle based arguments and/or analysis of the parameter trajectory is employed. An alternative to such analysis, albeit approximate, is to use the well-known Describing Function (DF) method. In doing so, the analysis of the robust adaptive control problem becomes tractable and resembles that of familiar and classical linear stability analysis. Thesis Supervisor: Dr. Anuradha M. Annaswamy Title: Senior Research Scientist

A relay feedback structure for processes under static disturbances or drift

Abstract: The standard relay feedback method cannot provide a stable oscillation under large static disturbances or drift. In this paper, it is proposed a relay feedback structure to overcome this limitation. This structure is composed by a block to remove static disturbance or drift followed by a relay. The block consists of a simple high-pass filter followed by a relay plus an integrator. Describing function analysis shows that the proposed structure is similar to the standard relay. Case studies illustrate that the proposed relay structure results in a symmetric oscillation of the process output.

Method of Harmonic Balance in Full-Scale-Model Tests of Electrical Devices

Abstract: Methods for determining the weber-ampere characteristics of electrical devices, one of which is based on solution of direct problem of harmonic balance and the other on solution of inverse problem of harmonic balance by the method of full-scale-model tests, are suggested. The mathematical model of the device is constructed using the describing function and simplex optimization methods. The presented results of experimental applications of the method show its efficiency. The advantage of the method is the possibility of application for nondestructive inspection of electrical devices in the processes of their production and operation.

Modelling and analysis of nonlinear thermoacoustic systems using frequency and time domain methods

Abstract: Thermoacoustic oscillations may arise in combustion chambers when unsteady heat release and acoustic fluctuations constructively interfere. These oscillations generally lead to undesired consequences, and need to be avoided. Linear stability analysis can be used to investigate the linear stability of a thermoacoustic system, by calculating the frequencies and growth rates of thermoacoustic modes. Adjoint methods can then be used to understand what parameters in the configuration under investigation have to be changed to make it less susceptible to thermoacoustic oscillations. Linear stability is, however, not sufficient in general to ensure safe operability conditions. This is because nonlinear and non-normal effects may trigger finite amplitude oscillations when the system is subject to finite amplitude perturbations. A thorough fully nonlinear investigation of thermoacoustic systems is prohibitively expensive both experimentally and numerically, and one needs to approximate the nonlinear response of the system. In this thesis, low-order nonlinear models for the prediction of the nonlinear behaviour of thermoacoustic systems are developed. These models are based on thermoacoustic networks, in which linear acoustics is combined with a nonlinear heat release model. The acoustic networks considered in this thesis can take into account mean flow and non-trivial acoustic reflection coefficients, and are cast in state-space form to enable analysis both in the frequency and time domains. Starting from linear analysis, the stability of thermoacoustic networks is investigated, and the use of adjoint methods for understanding the role of the system’s parameters on its stability is demonstrated. Then, a fully nonlinear analysis using various state-of-the-art methods is performed, to highlight the strengths and weaknesses of each method. Two novel frameworks that fill some gaps in the available methods are developed: the first, called Flame Double Input Describing Function (FDIDF), is an extension of the Flame Describing Function (FDF). The FDIDF approximates the flame nonlinear response when it is forced simultaneously with two frequencies, whereas the FDF is limited to one frequency. Although more expensive to obtain, the FDIDF contains more nonlinear information than the FDF, and can predict periodic and quasiperiodic oscillations. It is shown how, in some cases, it corrects the prediction of the FDF about the stability of thermoacoustic oscillations. The

Frequency shaped sliding mode control of magnetorheological smart structure systems

Abstract: This paper addresses the problem of controlling multi-degree-of-freedom (MDoF) smart structures integrated with magnetorheological (MR) devices that are subject to non-linearity and hysteresis. A fluid based device, namely the MR damper (MRD), is considered in this study, where hysteresis appears in both force-displacement and force-velocity relationships of the smart device. Such nonlinear dynamics limit the performance of the device when embedded in smart structures. The describing function (DF) technique is employed using only the displacement as input to the nonlinearity to characterize this multivalued mechanism. By incorporating the proposed model into the system dynamics, frequency shaped sliding mode control (FSSMC) is developed to achieve structural resilience and sustainability against nonlinearities, modeling uncertainties, and disturbances from dynamic loadings. Frequency response functions (FRFs) are obtained for possible analysis of system conditional assessment in the frequency domain. Simulations are reported for a three-story building model integrated with two identical current-dependent MR dampers subject to one-dimensional quake-induced vibration to investigate lateral dynamic responses, as produced by earthquakes or strong winds.