Spectral Analysis of Block Structured Nonlinear Systems

A systematic frequency domain method for nonlinear analysis, design and estimation of nonlinear systems is established based on the discussions in the previous chapters. Firstly, this method allows accurate determination of the linear and nonlinear components in system output spectrum of a given nonlinear system described by NDE, NARX or NBO (nonlinear block-oriented) models, with some simulation or experiment data. These output spectrum components can then be used for system identification or nonlinear analysis for different purposes such as fault detection etc. Secondly, the OFRF discussed before is expressed into a much improved polynomial function, referred to here as nonlinear characteristic output spectrum (nCOS) function, which is an explicit expression for the relationship between nonlinear output spectrum and system characteristic parameters of interest including nonlinear parameters, frequency variable, and input excitation magnitude (not just nonlinear parameters as that in Chaps. 7 and 8 ) with a more generic parametric structure. With the accurate determination of system output spectrum components in the previous step, the nCOS function can therefore be accurately determined up to any high orders, with less simulation trials and computation cost compared with a pure simulation based study or traditional theoretical computation. These results can provide a useful approach for qualitative and quantitative analysis and design of nonlinear dynamics in the frequency domain.

Purpose – Analytical determination of harmonic components of current in electric circuits containing semiconductor converters with the use of a small parameter method (SPM) in frequency domain. The paper aims to discuss these issues. Design/methodology/approach – A SPM realized in frequency domain was used in the analytical analysis of electric circuits with semiconductor converters. An automated method of formation of orthogonal harmonic components of electrical values on the basis of discrete convolution algorithm was used to provide the possibility of realization of calculation in frequency domain. A nonlinear characteristic of a semiconductor converter was presented by the method of numerical approximation. A numerical structured simulation method was applied to determination of the reference values of current in the analyzed circuit. Laws of theoretical electrical engineering were used for formation of the equations of voltage balance in the circuit with a nonlinear element. Findings – It is shown th...

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

We show how frequency domain design procedures can be effectively used in nonlinear fuzzy control systems. For this purpose, theoretical justifications of the approach discussed here are provided to show that solid notions from nonlinear system and control theory can be successfully applied in the fuzzy control domain. The methodology proposed is very simple, but powerful, effective and practical. In addition, it supports systematic control system design and stability analysis in the frequency domain for systems described by mathematical, linguistic or experimental models. Several case studies are included to illustrate the usefulness of frequency response in conventional fuzzy control systems analysis and design.

The high penetration of power electronic converters with complex control systems is changing the power system dynamics, introducing new challenges such as converter-driven stability incidents. Traditional stability analysis methods, suitable for classical problems like voltage, frequency, and rotor angle stability in large systems, are insufficient for addressing the fast control dynamics of converters, which involve electromagnetic phenomena. These phenomena require detailed converter and network modeling, which can be performed in both the frequency and time domains, enabling the respective stability analyses to be carried out. However, frequency domain methods, based on small-signal impedances linearized at a single operating point, inherently ignore time domain phenomena like switching events and nonlinear behaviors. In contrast, time domain electromagnetic transient (EMT) simulations are effective for analyzing converter-driven stability but are computationally intensive when applied to large transmission systems with numerous use cases. Therefore, to reduce the simulation complexity in EMT tools, a complexity reduction procedure is proposed in this paper. Leveraging the advantages of the frequency domain, such as faster simulation times and information on wideband frequency characteristics of the system, this procedure utilizes the small-signal impedances and introduces a method for network reduction. The procedure also uses the frequency domain stability analysis method to screen for critical network use cases. Primarily, this procedure is a frequency domain toolchain encompassing frequency domain stability analysis and frequency domain network reduction. The result of the toolchain is a reduced network size and reduced network use cases that can be used for EMT simulations. The procedure is applied to an IEEE 39 bus system, where converter-driven stability is evaluated for two use cases. Furthermore, the network reduction method is tested on a critical use case, demonstrating reductions in network size and computation times without compromising the quality of stability analysis results.

The analysis of multidegree-of-freedom (MDOF) nonlinear systems is studied using the concept of nonlinear output frequency response functions (NOFRFs), which are a series of one-dimensional functions of frequency recently proposed by the authors to facilitate the analysis of nonlinear systems in the frequency domain. By inspecting the relationships between the NOFRFs of two consecutive masses in the locally nonlinear MDOF systems, a series of properties about the NOFRFs of the nonlinear systems are obtained. These properties clearly reveal how the system linear characteristic parameters govern the distribution of nonlinear effects induced by the system nonlinear component within the whole systems. The results obtained in the present work are of significance to the frequency response analysis for nonlinear systems. One important potential application of these results is the detection and location of damages in engineering structures, which make the structures behave nonlinearly.

Compensation and control of nonlinear systems in frequency domain

Vehicle suspension (or vibration control) systems are usually inherently nonlinear and can be modeled as single input multiple output (SIMO) system. In this paper, parametric convergence bounds for Volterra series expansion of nonlinear systems described by a SIMO nonlinear auto-regressive with exogenous inputs model are studied in the frequency domain, which can clearly indicate the parametric range in which a given nonlinear system has a convergent Volterra series expansion, referred to as parametric bound of convergence (PBoC). With the resulting PBoC of characteristic parameters, nonlinear systems with a nonlinear multiobjective performance (MOP) function can then be analyzed in the frequency domain using a nonlinear characteristic output spectrum method based on the Volterra series expansion. To demonstrate the results and method above, a vehicle suspension system, which is taken as a typical SIMO nonlinear system with a MOP function to optimize, is investigated. The results demonstrate a systematic and novel method for nonlinear analysis and design.

This paper presents the analytical modeling, modal testing, and finite-element model updating for a two-span masonry arch bridge. An Ottoman masonry arch bridge built in the 19th century and located at Camlihemsin, Rize, Turkey is selected as an example. Analytical modal analysis is performed on the developed 3D finite-element model of the bridge to obtain dynamic characteristics. The ambient vibration tests are conducted under natural excitation such as human walking. The operational modal analysis is carried out using peak picking method in the frequency domain and stochastic subspace identification method in the time domain, and dynamic characteristics (natural frequencies, mode shapes, and damping ratios) are determined experimentally. Finite-element model of the bridge is updated to minimize the differences between analytically and experimentally estimated dynamic characteristics by changing boundary conditions. At the end of the study, maximum differences in the natural frequencies are reduced on average from 18 to 7% and a good agreement is found between analytical and experimental dynamic characteristics after finite-element model updating.

Non-linear output frequency response functions of MDOF systems with multiple non-linear components

This paper is concerned with distributed consensus control of unknown multiagent systems. As the system’s dynamics is unknown, an adaptive Fourier decomposition (AFD) based iterative learning control (ILC) dynamics adaptive matching method in frequency domain is put forward to deal with it. First, large amounts of input and output measurement data are used to estimate the frequency domain characteristics of the system by Takenaka-Malmquist functions. Second, convert the traditional time domain ILC to the frequency domain to establish a matching relationship with the estimated frequency domain features. Then, an adaptive iterative learning rate is constructed to achieve the optimal convergence at each sampling point. The feasibility of the proposed algorithm is guaranteed by the convergence of AFD in Hardy space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H^{2}(\mathbb {D})$ </tex-math></inline-formula> under the maximum selection principle. Compared with the reinforcement learning data-driven control scheme, the method proposed in this paper has obvious advantages in the control accuracy and convergence efficiency. In addition, this paper takes two kinds of denoising algorithms based on unwinding AFD to deal with the multi-agent systems with channel noise. Finally, the feasibility and effectiveness of the developed method are verified by a series of simulations.

The describing function approach is a powerful tool for characterizing nonlinear dynamical systems in the frequency domain. In this paper, we extend the describing function approach to detect and localize the damage in initially healthy nonlinear systems with limited measurements. The requirement of complete FRF of the underlying linear system by describing function approach is overcome by using a newly developed nonparametric principal component analysis-based model. Numerical simulation studies have been carried out by considering a cantilever beam with multiple local nonlinear attachments to demonstrate the localization process of the improved describing function approach with limited instrumentation. Parametric estimation of a shear building model is considered as a second numerical example to demonstrate the capability of the proposed approach in identifying the different types of nonlinearities and as well as combined types of nonlinearities (i.e. more than one type of nonlinearity). These combined nonlinearities can exist either in the same or different spatial locations. Experimental investigations have also been presented in this paper to complement the numerical investigations to demonstrate the practical applicability.

One of the most versatile and powerful approaches to the identification of nonlinear dynamical systems is the NARMAX (Nonlinear Auto_regressive Moving Average with eXogenous inputs) method. The model represents the current output of a system by a nonlinear regression on past inputs and outputs and can also incorporate a nonlinear noise model in the most general case. Although the NARMAX model is most often given a polynomial form, this is not a restriction of the method and other formulations have been proposed based on nonparametric machine learning paradigms, for example. All of these forms of the NARMAX model allow the computation of Higher-order Frequency Response Functions (HFRFs) which encode the model in the frequency domain and allow a direct interpretation of how frequencies interact in the nonlinear system under study. Recently, a NARX (no noise model) formulation based on Gaussian Process (GP) regression has been developed. One advantage of the GP NARX form is that confidence intervals are a natural part of the prediction process. The objective of the current paper is to discuss the GP formulation and show how to compute the HFRFS corresponding to GP NARX. Examples will be given based on simulated data.KeywordsNonlinear system identificationNARMAX modelsHigher-order Frequency Response Functions (HFRFs)Gaussian processes

Clearance turns up in a large number of engineering structures because of the errors during assembling, manufacturing, and wearing. The presence of clearance in engineering structures changes the normal dynamic response and will result in low precision and short lifetime. The clearance parameter identification of such nonlinear system is the prerequisite to control and eliminate the effect of clearance nonlinearity. In this paper, a derivative plot of probability density function (DPPDF) for displacement response is proposed to precisely identify the clearance value of continuous system, and the nonlinear subspace identification (NSI) method is modified to recognize the related contact stiffness based on the frequency response function (FRF) equations of continuous system. The DPPDF method is carried out by analyzing the distribution characteristic of displacement response, and the clearance value is derived through inspecting the probability density function (PDF) plot and the second derivative plot of the PDF. Based on the identified clearance, the clearance nonlinearity is regarded as external force, and the relationship between the dynamic responses and the external forces in frequency domain can be expressed as the form of FRF equations. Based on the FRF equations, the contact stiffness in continuous system is obtained with modified NSI method. This combined identification process is verified by a single-degree-of-freedom (SDOF) system and a cantilever beam system with clearances, and some influence factors of this identification process, including noise, transfer error, and force level, are discussed in detail. In the end, an experiment device with changeable clearance and contact stiffness was designed to conduct identification experiments, and the results show that the proposed methods perform effectively in identifying the clearance parameters.

The case study presents modern robust control methods of the Bergman minimal model of Type I diabetic patients under intensive care using computer algebra: the disturbance rejection LQ control or minimax control (as extension of the classical LQ control) and the robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sub> control. It is shown the minimax control has limitations in practice, but employing reduced Gro¿bner basis on rational field, it is possible to approximate the theoretical solution and so getting a better solution than LQ does. The symbolic and numeric computations were carried out with Mathematica 5.2 and with its CSPS Application, as well as with MATLAB 6.5. In case of H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sub> control, the graphical method in frequency domain- implemented under Mathematica by-is extended with a disturbance rejection constraint and the robustness of the resulted highorder linear controller is demonstrated by nonlinear closed loop simulation in state-space, in case of standard meal disturbances.