Nonlinear Systems Theory Research Articles

Analysis of Lur’e dominant systems in the frequency domain

Frequency domain analysis of linear time-invariant (LTI) systems in feedback with static nonlinearities is a classical and fruitful topic of nonlinear systems theory. We generalize this approach beyond equilibrium stability analysis with the aim of characterizing feedback systems whose asymptotic behavior is low dimensional. We illustrate the theory with a generalization of the circle criterion for the analysis of multistable and oscillatory Lur’e feedback systems.

Novel Homotopy Theory for Nonlinear Networks and Systems and Its Applications to Electrical Grids

Homotopy methods have several useful applications in biology, economics, nonlinear circuit design, and electric power networks, among others. A novel homotopy theory and a convergence theorem are presented in this paper for general-purpose homotopy methods. It is shown that the derived analytical results are applicable to a class of large-scale integrated-circuit designs and protein interaction networks. In addition, an application of this theory is illustrated to prove the convergence of the homotopy-enhanced implicit Z-bus power-flow method. The existence of power-flow solutions for general distribution networks with distributed generators modeled as photovoltaic nodes is also justified. Numerical studies on the IEEE 37-bus, 123-bus, and 8500-node distribution networks are conducted to illustrate the set of sufficient conditions for the main theorem.

Nonlinear Power System Analysis Using Koopman Mode Decomposition and Perturbation Theory

A new model-based framework for studying nonlinear dynamic behavior of stressed power systems that combines Koopman mode analysis and perturbation theory is proposed. First, a systematic approach to deriving high-dimensional representations of nonlinear dynamical systems based on perturbation theory is presented. Nonlinear behavior is interpreted as a projection of the Koopman operator eigenfunctions of an extended coordinate system onto the physical variables of the system. Analytical formulations are then obtained for the construction of the extended coordinate system based on perturbation theory. The analysis methodology is demonstrated on three test systems: A single-machine, infinite-bus system, a three-machine test system, and the IEEE 50-machine test system.

A Fault Detection Approach for Nonlinear Systems Based on Data-Driven Realizations of Fuzzy Kernel Representations

This paper is devoted to the data-driven fault detection of nonlinear systems. For our purpose, the definition of Takagi–Sugeno fuzzy data-driven forms of kernel representations for nonlinear systems is introduced first, which builds the basis of our work. The major contributions consist of two parts. In the first part, a data-driven method for fuzzy process modeling is proposed, and associated with it, some modeling issues are addressed with the aid of the so-called randomized algorithm technique in the probabilistic framework. It is followed by a data-driven realization of fuzzy kernel representation and its implementation in the fault detection system design. To link the data-driven methods to the well-established observer-based fault detection approaches, the recursive form of the fuzzy kernel representation is proposed. In the second part, the fuzzy-observer-based fault detection design scheme is investigated based on the recursive fuzzy kernel representation. The main results of our study are illustrated by an experimental study on the laboratory setup of a three-tank system.

Dynamics and implications on a cooperative advertising model in the supply chain

In this paper, the theory of nonlinear dynamic system is applied to a cooperative advertising model in the supply chain, where the decision-making mechanism of firms is modeled as a dynamic adjustment process and the advertising response is assumed to be an exponential function. We find four possible Nash equilibrium points in this dynamic system and provide the conditions for their existence. After that, we analyze the impacts of key parameters on the stability of the positive Nash equilibrium point and prove that two kinds of bifurcation may occur when this positive Nash equilibrium point becomes unstable. Finally, sensitivity analysis and robustness test are performed to obtain some managerial insights. This study provides some valuable insights both into the cooperative advertising programs and into the features of dynamic game with exponential demand function.

Affine nonlinear control for an ultra-supercritical coal fired once-through boiler-turbine unit

The ultra-supercritical once-through boiler (OTB) unit is an advanced power generation technology with high plant efficiency, high coal utilization, and low emission. However, it is difficult to realize a coordinate control for the ultra-supercritical OTB unit to achieve fast and stable dynamic response during load tracking and grid frequency disturbances, and its complexity and nonlinearity result in the ultra-supercritical OTB unit becoming a multiple-input and multiple-output system filled with severe coupling and interference. According to the affine nonlinear system theory, a nonlinear control method is developed to solve nonlinear decoupled optimal control problems of a nonlinear ultra-supercritical OTB unit. Firstly, the exact linearization technique is used to transform the nonlinear ultra-supercritical OTB system into a linear one and an optimal LQR is designed for the corresponding nominal system. The optimal nonlinear control law can be derived by coordinate conversion and condition feedback and the study can be conducted for reasonable experiments and analysis. Then, compared with the linear PID control and model predictive control, the effectiveness and superiority of the proposed nonlinear optimal control have been verified through different loads disturbances from 50% to 100% using the Matlab/Simulink Toolbox.

Nonlinear Characteristics of Electrocardiograph Signals Based on Fractal

In view of the nonlinear properties of Electrocardiograph (ECG) signal, the application of fractal methods from nonlinear system theory for the analysis of ECG signals has gained increasing interest.In this study, analysis of the objects are ECG signals of four sinus rhythms. Some important phenomena and conclusions have been captured and drawn after analyzing with and plotting the graphics of multi-fractal spectrum and auto-correlation functions. Additionally, the Hurst(H) parameters illustrate that self-similarity is a common property of the ECG signals, but the smaller H of the normal sinus rhythm(NSR) cause the obvious randomness of NSR. The further research of multi-fractal spectrum shows that the ECG signals all present local singular characteristics, but there are inconsistencies in the same type of sinus rhythm ECG signal. While, the inconsistency led to obvious classification, especially in NSR. As the conclusion, the results can be used as an effective complementary method for non-invasive diagnosis and early warning of heart disease.

Detecting multi-pulse chaotic dynamics of high-dimensional non-autonomous nonlinear system for circular mesh antenna

This paper investigates the global bifurcations and multi-pulse jumping chaotic dynamics of circular mesh antenna. An equivalent continuum circular cylindrical shell is employed to represent the circular mesh antenna. Based on the four-dimension non-autonomous nonlinear governing equations of motion for the equivalent continuum circular cylindrical shell derived by Zhang et al. (2016, 2017), the improved extended Melnikov theory of the non-autonomous nonlinear system is utilized to conduct a theoretical analysis of the multi-pulse jumping chaotic motions for the equivalent continuum circular cylindrical shell. The thermal excitation and damping coefficient are considered as the controlling parameters to analyze their effect on the nonlinear vibrations and bifurcations of the equivalent continuum circular cylindrical shell. Numerical simulations are also introduced to further verify the existence of the multi-pulse jumping chaotic motions for the equivalent continuum circular cylindrical shell. The results obtained from the numerical simulations are compared to those obtained from the Melnikov theoretical prediction.

Two novel static and dynamic voltage stability based indexes for power system contingency ranking

The goal of contingency ranking is to accurately choose a list of critical contingencies and rank them due to their severity. In this study, two novel and effective approaches are presented for contingency ranking. The first method proposes a static voltage stability based index which is calculated from the modal analysis and the branch participation matrix. Based on average participation factor for each line, the average branch influence factor is calculated and then root sum square index is applied to rank the contingencies. The second approach is based on dynamic voltage stability. The method employs the largest Lyapunov exponent, a stability tool adapted from the theory of chaos and non-linear dynamical system, for time series. Time-series data of the weakest bus voltage are employed for the largest Lyapunov exponent computing and based on this quantity, contingency ranking will be carried out. Test results on the IEEE 14 bus system are presented and compared with three previous indices to show the effectiveness of both methods.

Geometric Theory of Reduction of Nonlinear Control Systems

The foundations of a differential geometric theory of nonlinear control systems are described on the basis of categorical concepts (isomorphism, factorization, restrictions) by analogy with classical mathematical theories (of linear spaces, groups, etc.).

The theory of nonlinear systems as an instrument for solving engineering problems

The article outlines theoretical, methodological and practical issues of modern control and optimization theory, as well as the problems of nonlinear systems theory. Theoretical conclusions and results allowed to build mathematical models applicable to the management of objects of different nature with different principles of action, in particular, to the management of complex technical and technological objects that can be considered as nonlinear dynamic systems. The authors find it appropriate to consider nonlinear dynamic integral models as Volterra integro-power series from many functional arguments with multidimensional weight functions and a certain finite set of inputs to the system. The set of multidimensional kernels of integral Volterra operators completely characterizes the nonlinear and dynamic properties, and, consequently, the technical state of the initial system. The application of Volterra series based models allows to take into account the nonlinear and inertial properties of the initial nonlinear dynamic system more fully and accurately, it also makes the model diagnostic of a technical system more universal, raises the reliability of the forecast. The diagnostic procedure in this case is aimed at defining Volterra kernels based on the data of “input-output” experiment and building the diagnostic system of attribute in the space of which the decisive rule of optimal classification is created.

Reduced linear fractional representation of nonlinear systems for stability analysis

Based on symbolic and numeric manipulations, a model simplification technique is proposed in this paper for the linear fractional representation (LFR) and for the differential algebraic representation introduced by Trofino and Dezuo (2013). This representation is needed for computational Lyapunov stability analysis of uncertain rational nonlinear systems. The structure of the parameterized rational Lyapunov function is generated from the linear fractional representation (LFR) of the system model. The developed method is briefly compared to the n-D order reduction technique known from the literature. The proposed model transformations does not affect the structure of Lyapunov function candidate, preserves the well-posedness of the LFR and guarantees that the resulting uncertainty block is at most the same dimensional as the initial one. The applicability of the proposed method is illustrated on two examples.

Phase Shadows: An Enhanced Representation of Nonlinear Dynamic Systems

Many nonlinear dynamic systems have a rotating behavior where an angle defining its state may extend to more than 360[Formula: see text]. In these cases the use of the phase portrait does not properly depict the system’s evolution. Normalized phase portraits or cylindrical phase portraits have been extensively used to overcome the original phase portrait’s disadvantages. In this research a new graphic representation is introduced: the phase shadow. Its use clearly reveals the system behavior while overcoming the drawback of the existing plots. Through the paper the method to obtain the graphic is stated. Additionally, to show the phase shadow’s expressiveness, a rotating pendulum is considered. The work exposes that the new graph is an enhanced representational tool for systems having equilibrium points, limit cycles, chaotic attractors and/or bifurcations.

Computing Sparse GCD of Multivariate Polynomials via Polynomial Interpolation

The problem of computing the greatest common divisor (GCD) of multivariate polynomials, as one of the most important tasks of computer algebra and symbolic computation in more general scope, has been studied extensively since the beginning of the interdisciplinary of mathematics with computer science. For many real applications such as digital image restoration and enhancement, robust control theory of nonlinear systems, L1-norm convex optimization in compressed sensing techniques, as well as algebraic decoding of Reed-Solomon and BCH codes, the concept of sparse GCD plays a core role where only the greatest common divisors with much fewer terms than the original polynomials are of interest due to the nature of problems or data structures. This paper presents two methods via multivariate polynomial interpolation which are based on the variation of Zippel’s method and Ben-Or/Tiwari algorithm, respectively. To reduce computational complexity, probabilistic techniques and randomization are employed to deal with univariate GCD computation and univariate polynomial interpolation. The authors demonstrate the practical performance of our algorithms on a significant body of examples. The implemented experiment illustrates that our algorithms are efficient for a quite wide range of input.

Complexity-Entropy Maps as a Tool for the Characterization of the Clinical Electrophysiological Evolution of Patients under Pharmacological Treatment with Psychotropic Drugs

In the clinical electrophysiological practice, reading and comparing electroencephalographic (EEG) recordings are sometimes insufficient and take too much time. Tools coming from the information theory or nonlinear systems theory such as entropy and complexity have been presented as an alternative to address this problem. In this work, we introduce a novel method—the permutation Lempel–Ziv Complexity vs. Permutation Entropy map. We apply this method to the EEGs of two patients with specific diagnosed pathologies during respective follow up processes of pharmacological changes in order to detect alterations that are not evident with the usual inspection method. The method allows for comparing between different states of the patients’ treatment, with a healthy control group, given global information about the signal, supplementing the traditional method of visual inspection of EEG.

Nonlinear signal processing in systems with random structure for the case of spatial-time-varying colored Gaussian-Markov noise

The extension of the adaptive estimation theory for nonlinear correlation-extremum systems with random structure to the case when the measurements noise model is a spatial-time-varying Gaussian-Markov colored process is presented and the new signal processing algorithms are derived which provide the system operation in varying and uncertain external conditions.

Exponential stability and stabilization of extended linearizations via continuous updates of Riccati‐based feedback

SummaryMany recent works on the stabilization of nonlinear systems target the case of locally stabilizing an unstable steady‐state solution against small perturbations. In this work, we explicitly address the goal of driving a system into a nonattractive steady state starting from a well‐developed state for which the linearization‐based local approaches will not work. Considering extended linearizations or state‐dependent coefficient representations of nonlinear systems, we develop sufficient conditions for the stability of solution trajectories. We find that if the coefficient matrix is uniformly stable in a sufficiently large neighborhood of the current state, then the state will eventually decay. On the basis of these analytical results, we propose a scheme that is designed to maintain the stabilization property of a Riccati‐based feedback constant during a certain period of the state evolution. We illustrate the general applicability of the resulting algorithm for setpoint stabilization of nonlinear autonomous systems and its numerical efficiency in 2 examples.

Finite Dimensional State Representation of Linear and Nonlinear Delay Systems

We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear autonomous systems and linear systems with time-dependent input we give necessary and sufficient conditions and in the nonlinear case we give sufficient conditions. Most of our results for linear renewal and delay differential equations are known in different guises. The novelty lies in the approach which is tailored for applications to models of physiologically structured populations. Our results on linear systems with input and nonlinear systems are new.

Stability analysis of 2D Roesser systems via vector Lyapunov functions

The paper gives new results that contribute to the development of a stability theory for 2D nonlinear discrete and differential systems described by a state-space model of the Roesser form using an extension of Lyapunov’s method. One of the main difficulties in using such an approach is that the full derivative or its discrete counterpart along the trajectories cannot be obtained without explicitly finding the solution of the system under consideration. This has led to the use of a vector Lyapunov function and its divergence or its discrete counterpart along the system trajectories. Using this approach, new conditions for asymptotic stability are derived in terms of the properties of two vector Lyapunov functions. The properties of asymptotic stability in the horizontal and vertical dynamics, respectively, are introduced and analyzed. This new properties arise naturally for repetitive processes where one of the two independent variables is defined over a finite interval. Sufficient conditions for exponential stability in terms of the properties of one vector Lyapunov function are also given as a natural follow on from the asymptotic stability analysis.

Model Reduction by Differential Balancing Based on Nonlinear Hankel Operators

In this paper, we construct balancing theory for nonlinear systems in the contraction framework. First, we define two novel controllability and observability functions via prolonged systems. We analyze their properties in relation to controllability and observability, and use them for so-called differential balancing, and its application to model order reduction. One of the main contribution of this paper is showing that differential balancing has close relationships with the Frechet derivative of the nonlinear Hankel operator. Inspired by [3] , we provide a generalization in order to have a computationally more feasible method. Moreover, error bounds for model reduction by generalized balancing are provided.