Analysis Of Nonlinear Systems Research Articles

Balance on different unstable supports: a complementary approach based on linear and non-linear analyses.

Maintenance of postural control is a complex task that requires the integration of different sensory-motor processes. To improve postural control, balance training is often implemented using unstable surfaces. Little is known, however, about how different surfaces compare in terms of postural control strategy. Non-linear dynamical system analysis, like recurrent quantification analysis (RQA) applied to the center of pressure (CoP) trajectory, represents a useful tool in this respect. The aim of this study is to investigate the effects of different unstable supports on the CoP trajectory through a complementary approach based on linear and non-linear analyses. Seventeen healthy adults performed barefoot single-leg balance trials on a force plate and on three different balance training devices (soft disc, foam pad, and pillow). Sets of parameters were extracted from the CoP trajectories using classical stabilometric analysis (sway path, mean velocity, root mean square) and RQA (percent recurrence and determinism, maximum line length, entropy). Both classical and RQA analyses highlighted significant differences between stable (force plate) and unstable conditions (p < 0.001). Conversely, only classical stabilometric parameters showed significant differences among the considered balance training devices, indicating that the different characteristics of the devices do not influence the dynamic/temporal structure of the CoP trajectory. Analysis of the center of pressure trajectory during single-leg standing on three different balance training devices and on a rigid surface using both linear and non-linear techniques.

Modeling of Tumor Growth with Input from Patient-Specific Metabolomic Data.

Advances in omic technologies have provided insight into cancer progression and treatment response. However, the nonlinear characteristics of cancer growth present a challenge to bridge from the molecular- to the tissue-scale, as tumor behavior cannot be encapsulated by the sum of the individual molecular details gleaned experimentally. Mathematical modeling and computational simulation have been traditionally employed to facilitate analysis of nonlinear systems. In this study, for the first time tumor metabolomic data are linked via mathematical modeling to the tumor tissue-scale behavior, showing the capability to mechanistically simulate cancer progression personalized to omic information obtainable from patient tumor core biopsy analysis. Generally, a higher degree of metabolic dysregulation has been correlated with more aggressive tumor behavior. Accordingly, key parameters influenced by metabolomic data in this model include tumor proliferation, vascularization, aggressiveness, lactic acid production, monocyte infiltration and macrophage polarization, and drug effect. The model enables evaluating interactions of interest between these parameters which drive tumor growth based on the metabolomic data. The results show that the model can group patients consistently with the clinically observed outcomes of response/non-response to chemotherapy. This modeling approach provides a first step towards evaluation of tumor growth based on tumor-specific metabolomic data.

Quantifying periodic, multi-periodic, hidden and unstable regimes of a magnetic pendulum via semi-analytical, numerical and experimental methods

The dynamics of a magnetic pendulum mounted on a shaft with an elastic element is considered. The impulsive excitation of the pendulum is performed through the interaction of a strong neodymium magnet and an electric coil. The mathematical model of the pendulum is represented by the non-autonomous ordinary nonlinear differential equation. The model is verified in a wide range of motion regimes by the laboratory experiment. Periodic motions are considered in more detail by the semi-analytic approximate method utilizing the averaging ideas. The amplitude and phase shift of system solutions possess highly nonlinear features which we propose to approximate by the saw-tooth functions. This new approach allows one to improve the results of the application of the conventional averaging method and is promising for the application and analysis of nonlinear systems with a strong nonlinearity.

Performance analysis of nonlinear vibration energy harvesting system with inelastic barrier under colored noise excitation

The vibro-impact structures are introduced into the piezoelectric vibration energy harvesting (VEH) systems, which can broaden the operating bandwidth of the piezoelectric VEH systems and thus improve the harvesting efficiency. The process of vibro-impact not only causes energy loss during the impact process but also causes the local deformation between two impact bodies. Therefore, this manuscript proposes a class of stochastic non-classical inelastic impact VEH models, whose non-classical impact processes are described by Hertz contact forces. Firstly, the relationship between the total energy of the unperturbed system and the potential energy of the system at the barrier position is used to determine whether the impact phenomena have occurred or not, then used to derive the period and frequency of the system. Secondly, using the quasi-conservative stochastic averaging method, the averaged drift term and the averaged diffusion term of the piecewise smooth averaged Ito^ equation are derived. The stationary probability density functions (PDFs), the mean square output (MSO) voltage and the root mean square (RMS) voltage of the system response can be obtained from solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation for averaged Ito^ equation. Finally, an example is provided to verify the effectiveness of the proposed method, and the effects of the model parameters on the MSO voltage and power conversion efficiency (PCE) are analyzed. The relationship between the MSO voltage and the PCE is also analyzed. It is of great significance for the design and optimization of stochastic VEH systems.

Partial component consensus analysis of second-order and third-order nonlinear multi-agent systems

Partial component consensus is a phenomenon that some components of agents reach identity, but the others components are not the same. In this paper, the partial component consensus problem of second-order and third-order nonlinear multi-agent systems has been investigated. Some concepts and results of partial component consensus are given through partial component stability theory and pinning control protocol. Numerical simulations are finally given to verify the theoretical results.

Application of Nonlinear Correction Method for Attitude Control and Landing Oscillations Prevention

The purpose of this work is to demonstrate the application of the nonlinear correction method to prevent attitude oscillations of flying vehicles. This paper investigates the complications of manual control tasks in aeronautics caused by employing the integral component in the control loop. If the actuators of aircraft controlling surfaces reach their rate limits, the aircraft oscillations can appear for both manually and automatically piloted cases. In this paper, the oscillations alleviation problem is solved using the nonlinear correction method. A general methodology for pilot-induced oscillations prevention is described briefly by the example of an aircraft pitch control system. The obtained pilot model parameters are used for altitude control system examination. The nonlinear systems analysis is carried out by calculating the generalized sensitivity function for the control and disturbance input signal. In the second part, the application of the nonlinear correction method to prevent aircraft and satellite oscillations caused by integrator windup. A comparative analysis of the nonlinear correction and the customary anti-windup feedback is carried out. The results obtained demonstrate that a sequential nonlinear corrective device can be efficiently used to suppress pilot-induced and integrator windup oscillations.

Local Stabilization of Discrete-Time Nonlinear Systems

This article considers local stability analysis and local stabilization of discrete-time nonlinear systems represented by Takagi–Sugeno fuzzy models. Conditions are established using Lyapunov functions and controller gains that depend also on past samples. Together with the stability and design conditions, an estimate of the region of attraction is also determined. The developed conditions are discussed and illustrated on several examples.

On one approach to the qualitative analysis of nonlinear dynamical systems

On one approach to the qualitative analysis of nonlinear dynamical systems

On an Approach to Qualitative Analysis of Nonlinear Dynamic Systems

On an Approach to Qualitative Analysis of Nonlinear Dynamic Systems

Sea-Surface Small Floating Target Recurrence Plots FAC Classification Based on CNN

In this paper, we propose the False-Alarm-Rate-Controllable (FAC) classification of sea clutter Recurrence Plots (RPs) based on Convolutional Neural Networks (CNN), which is shortened as RPs-CNN. Sea clutter data is a non-linear and recursive time series, and RPs provide qualitative analysis of non-linear and recursive dynamic systems. Thus, we construct the RPs datasets to extract the recursive feature of sea clutter. In the RPs datasets, RPs parameters, embedding delay &#x03C4; and embedding dimension m, are obtained by the average mutual information (AMI) and false nearest neighbor (FNN) algorithms, respectively. In addition, in order to extract the local features of RPs, CNN is applied. CNN makes full use of local features of the datasets, has the advantages of translation invariance and strong generalization ability, and shares the available weights to simplify network structure. We implement a proper LeNet-5 CNN training on the constructed RPs datasets, and verity it by IPIX measured datasets. The experimental results demonstrate that the CNN successfully classifies the RPs of targets and clutter. Moreover, the feasibility of RPs-CNN is verified by seven aspects, i.e., Accuracy, Precision, False Alarm, Miss Rate, Recall, F1-measure and Kappa. In addition, six parameters that may affect the classification performance are also analyzed, including time-series length, embedding delay, embedding dimension, convolutional kernel size, kernel depth, and optimization function. The results indicate that the proposed RPs-CNN method can reach a 92.05% F1-measure and 87.19% Kappa which performs better than other classification methods. Meanwhile, the false alarm rate (FAR) of the RPs-CNN is testified by the FAC classification. Experimental results demonstrate that the proposed RPs-CNN significantly improves the detection probability over other classification detectors in low FAR cases.

Dynamic Modeling and Analysis of Nonlinear Compound Planetary System

In the vehicle composite planetary gear transmission system, nonlinear excitations such as time-varying meshing stiffness, backlash and comprehensive error would lead to large vibration and noise, uneven load distribution, unstable operation and other problems. To address these issues, this work focuses on compound planetary gears and develops the bending-torsion coupling nonlinear dynamic model of the system based on the Lagrange equation. There are internal and external multi-source excitations applied to the system. This model is used to study the bending-torsion coupling meshing deformation relationship of each meshing pair along with the translational and torsional directions. The natural frequencies and vibration modal characteristics of the system are extracted from the model, and the influence of rotational inertia and meshing stiffness on the inherent characteristics of the system are studied. The coupling vibration characteristics of the system under operating condition are analyzed in terms of the inherent characteristics and time–frequency characteristics of the system. The simulation results exhibit that the planetary gear system has three modes. The change in natural frequency trajectory has two phenomena: modal transition and trajectory intersection. The main frequencies include engine rotating frequency, meshing frequency and its double frequency, and the rotation frequency and harmonic frequency of the engine have a great influence on the vibration response of the system. Finally, the virtual prototype of the composite planetary system is used to verify the accuracy of the established model from speed, inherent characteristics, meshing force and frequency composition.

Computation of Koszul homology and application to involutivity of partial differential systems

The formal integrability of systems of partial differential equations plays a fundamental role in different analysis and synthesis problems for both linear and nonlinear differential control systems. Following Spencer's theory, to test the formal integrability of a system of partial differential equations, we must study when the symbol of the system, namely, the top-order part of the linearization of the system, is 2-acyclic or involutive, i.e., when certain Spencer cohomology groups vanish. Combining the fact that Spencer cohomology is dual to Koszul homology and symbolic computation methods, we show how to effectively compute the homology modules defined by the Koszul complex of a finitely presented module over a commutative polynomial ring. These results are implemented using the OREMORPHISMS package. We then use these results to effectively characterize 2-acyclicity and involutivity of the symbol of a linear system of partial differential equations. Finally, we show explicit computations on two standard examples.

On the incremental form of dissipativity

Following the seminal work of Zames, the input-output theory of the 70s acknowledged that incremental properties (e.g. incremental gain) are the relevant quantities to study in nonlinear feedback system analysis. Yet, non-incremental analysis has dominated the use of dissipativity theory in nonlinear control from the 80s. Results connecting dissipativity theory and incremental analysis are scattered and progress has been limited. This abstract investigates whether this theoretical gap is of fundamental nature and considers new avenues to circumvent it.

A parameter space method for analyzing Hopf bifurcation of fractional-order nonlinear systems with multiple-parameter

Hopf bifurcation analysis of fractional-order nonlinear systems with multiple-parameter is discussed in this paper. The regions and boundaries corresponding to Hopf bifurcation conditions are described in parameter space. Based on cylindrical algebraic decomposition, the parameter space is decomposed into finite number of connected regions by some boundaries. Then parameter space method is proposed to determine stable parameter region and Hopf bifurcation parameter hypersurface. One example illustrates the effectiveness of the method.

Averaging method and two-sided bounded solutions on the axis of systems with impulsive effects at non-fixed times

The averaging method, originally offered by Krylov and Bogolyubov for ordinary differential equations, is one of the most widespread and effective methods for the analysis of nonlinear dynamical systems. Further, the averaging method was developed and applied for investigating of various problems. Impulsive systems of differential equations supply as mathematical models of objects that, during their evolution, they are subjected to the action of short-term forces. Many researches have been devoted to non-fixed impulse problems. For these problems, the existence, stability, and other asymptotic properties of solutions were studied and boundary value problems for impulsive systems were considered. Questions of the existence of periodic and almost periodic solutions to impulsive systems also were examined. In this paper, the averaging method is used to study the existence of two-sided solutions bounding on the axis of impulse systems of differential equations with non-fixed times. It is shown that a one-sided, bounding, asymptotically stable solution to the averaged system generates a two-sided solution to the exact system. The closeness of the corresponding solutions of the exact and averaged systems both on finite and infinite time intervals is substantiated by the first and second theorems of N.N. Bogolyubov.

Forced vibration analysis of multi-degree-of-freedom nonlinear systems with the extended Galerkin method

In this study, the dynamic response behavior of a generalized nonlinear dynamic system is investigated using a newly proposed extended Galerkin method. The algebraic equations of vibration amplitudes are obtained through an integration of the weighted functions. The new method is equivalent to the harmonic balance method but with a much simpler calculation procedure and a higher efficiency. This is the first time to use the method for the analysis of nonlinear systems with high number of modes, manifesting that the method is applicable to forced vibrations of nonlinear behavior. The method is further validated by the numerical Runge-Kutta method.

Load step reduction for adjoint sensitivity analysis of finite strain elastoplasticity

In this paper, load step reduction techniques are investigated for adjoint sensitivity analysis of path-dependent nonlinear finite element systems. In particular, the focus is on finite strain elastoplasticity with typical hardening models. The aim is to reduce the computational cost in the adjoint sensitivity implementation. The adjoint sensitivity formulation is derived with the multiplicative decomposition of deformation gradient, which is applicable to finite strain elastoplasticity. Two properties of adjoint variables are investigated and theoretically proved under certain prerequisites. Based on these properties, load step reduction rules in the sensitivity analysis are discussed. The efficiency of the load step reduction and the applicability to isotropic hardening and kinematic hardening models are numerically demonstrated. Examples include a small-scale cantilever beam structure and a large-scale conrod structure under huge plastic deformations.

Fragility analysis of nonlinear soil-structure systems including foundation uplifting and soil yielding

Fragility analysis of nonlinear soil-structure systems including foundation uplifting and soil yielding

Security Analysis of Linearization for Nonlinear Networked Control Systems Under DoS

We explore a security analysis of nonlinear networked control systems under denial-of-service (DoS) attacks. In particular, we focus on the vulnerability of a linearization approach in a stabilization problem. When linearization-based control is used, DoS attacks can make the state leave the region of attraction. This situation can occur when the initial state lies outside a certain region around the equilibrium, the size of which depends on the strength of the attacks on the network. In this article, we first derive a sufficient condition for local stability of a nonlinear system under averagely constrained DoS attacks. Then, we investigate the region of attraction and derive an explicit relation between the initial states and the DoS parameters to guarantee convergence of the state trajectories. Furthermore, we investigate a destabilization problem from the perspective of malicious attackers and present an instability condition. By a numerical example, we demonstrate a new insight foattackers to induce undesired behaviors of the control system.

Power spectral density analysis for nonlinear systems based on Volterra series

A consequence of nonlinearities is a multi-harmonic response via a mono-harmonic excitation. A similar phenomenon also exists in random vibration. The power spectral density (PSD) analysis of random vibration for nonlinear systems is studied in this paper. The analytical formulation of output PSD subject to the zero-mean Gaussian random load is deduced by using the Volterra series expansion and the conception of generalized frequency response function (GFRF). For a class of nonlinear systems, the growing exponential method is used to determine the first 3rd-order GFRFs. The proposed approach is used to achieve the nonlinear system’s output PSD under a narrow-band stationary random input. The relationship between the peak of PSD and the parameters of the nonlinear system is discussed. By using the proposed method, the nonlinear characteristics of multi-band output via single-band input can be well predicted. The results reveal that changing nonlinear system parameters gives a one-of-a-kind change of the system’s output PSD. This paper provides a method for the research of random vibration prediction and control in real-world nonlinear systems.