Analysis Of Nonlinear Systems Research Articles

Efficient stability analysis approaches for nonlinear weakly-hard real-time control systems

This article considers the stability analysis of nonlinear control systems with failures in the feedback loop that satisfy weakly-hard real-time (WHRT) constraints. Such WHRT constraints are window-based guarantees like for example, that the feedback loop is at least closed for n times in each window of m sampling times. WHRT constraints can, e.g., describe the packet loss properties of a communication network. We consider two different actuation strategies to deal with failures in the feedback loop: holding the last received input or setting the input to zero if no new input arrives. We present stability analysis procedures for both strategies. The procedures are based on the emulation of a sampled-data controller from continuous-time feedback. For the hold strategy, we use a graph description for the WHRT constraints and a modified version of the Viterbi algorithm to bound a Lyapunov function candidate. For the zero strategy, we employ a bound on the evolution of the Lyapunov function candidate for zero input. For both strategies, efficient stability analysis procedures based on non-monotonic Lyapunov functions are derived. In addition, we demonstrate how switched controllers, that switch depending on the past dropout sequence, can be used for the hold strategy. Such switched controllers can significantly increase the maximum allowable sampling period for the hold strategy. The proposed approaches are illustrated with a numerical example.

Delay-dependent stability analysis of nonlinear truck-trailer systems based on non-fragile memory sampled-data via fuzzy control

Delay-dependent stability analysis of nonlinear truck-trailer systems based on non-fragile memory sampled-data via fuzzy control

Novel stability and passivity analysis for three types of nonlinear LRC circuits

In this paper, the global asymptotic stability and strict passivity of three types of nonlinear RLC circuits are investigated by utilizing the Lyapunov direct method. The stability conditions are obtained by constructing appropriate Lyapunov function, which demonstrates the practical application of the Lyapunov theory with a clear perspective. The meaning of Lyapunov functions is not clear by many specialists whose studies based on Lyapunov theory. They construct Lyapunov functions by using some properties of Lyapunov functions with much trial and errors or for a system choose candidate Lyapunov functions. So, for a given system Lyapunov function is not unique. But we insist that Lyapunov (energy) function is unique for a given physical system. In this study we highly simplified Lyapunov’s direct method with suitable tools. Our approach constructing energy function based on power-energy relationship that also enable us to take the derivative of integration of energy function. These aspects have not been addressed in the literature. This paper is an attempt towards filling this gap. The results are provided within and are of central importance for the analysis of nonlinear electrical, mechanical, and neural systems which based on the system energy perspective. The simulation results given from Matlab successfully verifies the theoretical predictions.

Novel convenient conditions for the stability of nonlinear incommensurate fractional-order difference systems

The absence of satisfactory results that address the stability analysis of nonlinear incommensurate Fractional-order Difference Systems (FoDSs) as well as the impossibility of establishing a proper Lyapunov function that helps us to reveal any stability outcome for those systems, are considered some significant nonlinear problems that should be taken into account. To overcome such dilemmas, the present work provides some novel convenient conditions that can be utilized to ensure the asymptotic stability of those systems. In particular, through utilizing some features of the Z-transform scheme, we intend to propose certain results that aim to reveal the stability of the nonlinear incommensurate fractional-order difference system formulated in the sense of the Caputo difference operator. These novel results are completely confirmed by demonstrating the stability of the numerical solutions for two nonlinear difference systems with incommensurate fractional-orders.

Application of Lyapunov–Floquet transformation to the nonlinear spacecraft relative motion with periodic-coefficients

The Lyapunov–Floquet (L–F) theory, a powerful result of Floquet theory, is one of the main tools for the stability analysis and study of nonlinear periodic systems. In this paper, Lyapunov–Floquet and modal transformations of the nonlinear approximation model of spacecraft relative motion, containing periodic state dependent coefficients (SDC), are developed. The transformations, advantageously, retain the stability characteristics of the original nonlinear relative motion dynamics. To provide a form to which averaging method can be applied, the nonlinear approximated equation of motion is scaled using the chief’s eccentricity. Through the application of L–F transformation, the linear part with periodic coefficients is reduced to time-invariant form and the transformed periodic nonlinear part has coefficients with true anomaly as the independent variable. The modal transformation reduced LF transformed equation into a form containing Jordan canonical form. Afterward, the averaging method is applied to the L–F transformed equation to describe the evolution of the motion in terms of the average values of the dynamical variables. Both the LF and modal transformed relative motion equations and the averaged equations, with chief in elliptical orbit, present forms that are useful and applicable for spacecraft guidance, navigation and control, formation flying, rendezvous and proximity operations.

Mathematical model analysis of nonlinear control systems of power turbines operating in condensation mode

The work is dedicated to modeling&turbine control systems and studying the stability of a nonlinearsystem. The dynamics of the turbine regulation system is described by a nonlinear&system of fourdifferential equations. This system of equations describes the mathematical model of a turbine operatingin condensing mode. Using the transformation&way, the mathematical model of the steamturbineregulation system is reduced to the conclusion of a problem about the unconditional stability of anonlinear stationary system of indirect control. In the study of a nonlinear system, the Lyapunov functionwas applied and the circumstances of unconditional stability were obtained. The conclusion of differentialequations systems is executed in vector-matrix form. The results&obtained for a nonlinear systemare&used to study the durability of a steam turbine. Numerical calculations are presented that illustratethe probability of using the proposed layout and studying the stability of nonlinear control systems forpower turbines. Key words: stability of a nonlinear system, differential equation, Lyapunov function, mathematicalmodel of a turbine, asymptotic stability, characteristic of nonlinearity.

An improved higher-order explicit time integration method with momentum corrector for linear and nonlinear dynamics

An improved explicit time integration method is proposed for linear and nonlinear dynamics. Its calculation procedure is obtained with cubic B-spline interpolation approximation and weighted residual method. In the formulation, a momentum corrector is used to improve actual computation accuracy, especially for some special discontinuous loads. Analytical solutions of the local truncation errors, algorithmic damping and period elongation have been deduced to obtain the influence of algorithmic parameters on these basis algorithmic properties. The proposed method possesses at least second-order accuracy and can achieve at most third-order accuracy for no physical damping case. With free algorithmic parameters, the proposed method has controllable stability and numerical dissipation. Some demonstrative numerical examples are tested to confirm high efficiency of the proposed method for a variety of dynamic problems such as, dynamic response analysis of linear systems under various representative applied loads, finite element analysis (FEA) for dynamic response of engineering structures, and nonlinear dynamic analysis for strong nonlinear system.

Resonance response analysis of nonlinear vibration energy harvesting system under bounded noise excitation

In this paper, the resonance response of piezoelectric vibration energy harvester (VEH) driven by bounded noise is discussed through the quasi-conservative stochastic averaging method. A nonlinear transformation based on the total energy is firstly established to transform piezoelectric VEH system from an electromechanical coupled nonlinear system into a single-degree-of-freedom (SDOF) system. Then the SDOF system is rewritten as It\^o stochastic system about the energy and residual phase under the case of p:q resonance through the quasi-conservative stochastic averaging method. And the joint probability density function (JPDF) of the stationary response is obtained by solving the corresponding two-dimensional Fokker-Planck-Kolmogorov (FPK) equation using the finite difference method. Meanwhile, the mean-square electric voltage and the mean output power are further analytically given through the JPDF. Finally, the resonance response of piezoelectric VEH system is analyzed in detail in case of the primary resonance, and the Monte Carlo (MC) simulation technique is adopted to validate the effectiveness of the finite difference method.

Response solutions of 3-dimensional degenerate quasi-periodic systems with small parameter

This paper considers a class of 3-dimensional real analytic nonlinear quasi-periodic systems with a small perturbation parameter, whose unperturbed part has a degenerate equilibrium point. Based on the Leray-Schauder Continuation Theorem [18] , [20] and using technique of outer parameter, we prove that the system has a small response solution for many sufficiently small parameters.

An Investigation into Accuracy, Computation, and Robustness of Stochastic Linearization of Systems with Saturation Nonlinearities

Quasilinear Control (QLC) is a theory with a set of tools that can be used for the analysis and design of stochastic feedback control systems with nonlinear actuators and sensors. It is based on Stochastic Linearization (SL), a method of linearization that approximates a nonlinearity with an equivalent gain and a bias using statistical properties of its input. While the QLC literature has studied many problems in controls (e.g., performance analysis and controller design), important issues such as accuracy, robustness, and computation of SL have not been fully investigated. This paper fills this gap, with a special focus on feedback systems with saturating actuators, which are commonly encountered in feedback control systems and the QLC literature. Specifically, the accuracy of SL in approximating the saturated response is investigated. A mathematical relationship is established between the saturation authority and the worst-case open-loop accuracy, and the implications on closed-loop accuracy are described. To study the robustness of SL in practical situations, the sensitivities of the equivalent SL gain and bias to system parameters are investigated. Finally, several root-finding algorithms are compared in terms of their computational speed in solving the SL equations, and a novel change of coordinates is presented to improve the performance. The results provide valuable insights into and methods for addressing the computational aspects of SL that are critical for proper design and analysis of nonlinear stochastic feedback control systems.

Controllability of non-linear fractional-order systems with damping behaviour and multiple delays

Abstract A controllability analysis of both linear and non-linear fractional-order systems with damping behaviour and multiple delays is studied. We derived the controllability result for damped system with multi-term fractional order and multiple delays by introducing some lemmas with the help of Laplace transform properties and Mittag–Leffler function. Further, some new sufficient conditions ensuring controllability for a class of non-linear multi-term fractional-order damped systems with multiple delays are established by utilizing fractional Caputo derivatives and Schauder’s fixed point theorem. Moreover, as applications that demonstrate the proposed results, two examples are presented.

Approximation of Linearized Systems to a Class of Nonlinear Systems Based on Dynamic Linearization

An alternative method to analyze a class of nonlinear systems in a bond graph approach is proposed. It is well known that the analysis and synthesis of nonlinear systems is not a simple task. Hence, a first step can be to linearize this nonlinear system on an operation point. A methodology to obtain linearization for consecutive points along a trajectory in the physical domain is proposed. This type of linearization determines a group of linearized systems, which is an approximation close enough to original nonlinear dynamic and in this paper is called dynamic linearization. Dynamic linearization through a lemma and a procedure is established. Therefore, linearized bond graph models can be considered symmetric with respect to nonlinear system models. The proposed methodology is applied to a DC motor as a case study. In order to show the effectiveness of the dynamic linearization, simulation results are shown.

Stability analysis of quasi one-sided Lipschitz non-linear multi-agent system via sampled data control subject to directed switching topology

Abstract This paper is concerned with the problem of stability and consensus of non-linear multi-agent system by utilizing the sampled-data control. The innovative part of this paper is that the nonlinearity of this class of nonlinear systems is considered to satisfy a quasi one-sided Lipschitz condition. Communication among agents are assumed to be a switching directed graph. The principle target of this paper is to design a sampled data controller such that for all permissible uncertainties, the resulting closed-loop system is stable in the sense of mean square. For this reason, through the development of an appropriate Lyapunov–Krasovskii functional with dual integral terms and usage of Kronecker product properties alongside the matrix inequality techniques, a new set of stability and consensus conditions for the prescribed system is obtained in the form of a linear matrix inequality, which can be easily solved by the well-known effective numerical programming. Finally numerical examples are given to show the validity of the proposed hypothetical results.

Time series analysis and prediction of nonlinear systems with ensemble learning framework applied to deep learning neural networks

In this paper, we design a framework to predict the value of time series for nonlinear systems. In order to achieve this goal, many studies of applications and plans for machine learning and even deep learning become currently popular. First, we select four nonlinear systems: including a proposed four-dimensional chaotic system, Lorenz system, Duffing oscillator, and Rössler attractor. The framework has three learning parts as Long Short-Term Memory (LSTM) based on Generate Performance Model (GPM), ensemble learning based on Restrict and Control Model (RCM), and one-dimensional convolution neural network (1-DCNN) of dirichlet distribution based on Overall Verification Model (OVM). Before learning steps, we exploit K-means method as pre-processing and hypothesis verification to improve the prediction accuracy. After learning steps, we construct four forecasting progresses as Point by Point Generated Method (PPGM), Sequence Full Generated Method (SFGM), Sequence Multiple Generated Method (SMGM), and Improvement with RCM and OVM (IPRO) to predict the value of the time steps. Finally, we use Mean Average Error (MAE) as the criterion of the prediction, and estimate the accuracy by comparing the error region of the average standard deviation.

An Algorithm for Output Spectrum Analysis of Nonlinear System with Correlative Sources Based on Power Series

An algorithm for output spectrum analysis of nonlinear system with correlative sources is proposed. First, the output terms of nonlinear system are analyzed based on the power series. Then, the algorithm is proposed. In the proposed method, different terms with the same frequency are picked out and added together by considering different phases, and the amplitudes of the output spectrum are obtained from the summarized results. Two numerical examples are used to verify the proposed algorithm. It is found that the amplitudes of output spectrum vary with different combinations of phases. Finally, the proposed algorithm is applied to control the output spectrum. Results show that the output amplitude of a specific frequency can be suppressed through changing both the amplitudes and the phases of the input signals.

Nonlinear model standardization for the analysis and design of nonlinear systems with multiple equilibria

In engineering practice, a nonlinear system stable about several equilibria is often studied by linearizing the system over a small range of operation around each of these equilibria, and allowing the study of the system using linear system methods. Theoretically, for operations beyond a small range but still within the stable regime of an equilibrium, the system behaves nonlinearly, and can be described and investigated using the Volterra series approach. However, there is still no available approach that can systematically transform the model of a nonlinear system into a form that can be studied over the whole stable regime about an equilibrium so as to facilitate the system study using the Volterra series approach. This transformation is, in the present study, referred to as nonlinear model standardization, which is the extension of the well-known concept of linearization to the nonlinear case. In this paper, a novel approach to nonlinear model standardization is proposed for nonlinear systems that can be described by a Nonlinear AutoRegressive model with eXogeneous input (NARX) or a nonlinear differential equation (NDE) model. The proposed approach is then used in three case studies covering the applications in nonlinear system analysis, nonlinear system design, and nonlinearity compensation, respectively, demonstrating the significance of the proposed nonlinear model standardization in a wide range of engineering practices.

Performance analysis of nonlinear spatial modulation multiple-input multiple-output systems

Despite the advantage of power efficiency and spectral efficiency, spatial modulation (SM) technology makes communication systems sensitive to the transmission impairment. This paper analytically investigates the performance of SM multiple-input multiple-output (MIMO) systems where power amplifier (PA) shows clear nonlinearity. Firstly, the system model is presented with nonlinear distortion included. Secondly, the system symbol error rate (SER) is newly derived to a polynomial based function of system parameters. Thirdly, the asymptotical behaviour of SER is analysed, and it is found that SER in nonlinear SM-MIMO systems has a lower limit. Finally, the analytical expressions of SER lower bond are presented. The analysis methodology and derived findings can be a useful reference for the designers of SM-MIMO systems.

Sparse Polynomial Zonotopes: A Novel Set Representation for Reachability Analysis

We introduce sparse polynomial zonotopes, a new set representation for formal verification of hybrid systems. Sparse polynomial zonotopes can represent nonconvex sets and are generalizations of zonotopes, polytopes, and Taylor models. Operations like Minkowski sum, quadratic mapping, and reduction of the representation size can be computed with polynomial complexity w.r.t. the dimension of the system. In particular, for reachability analysis of nonlinear systems, the wrapping effect is substantially reduced using sparse polynomial zonotopes, as demonstrated by numerical examples. In addition, we can significantly reduce the computation time compared to zonotopes when dealing with nonlinear dynamics.

Lyapunov stability analysis for nonlinear delay systems under random effects and stochastic perturbations with applications in finance and ecology

This manuscript is involved in the study of stability of the solutions of functional differential equations (FDEs) with random coefficients and/or stochastic terms. We focus on the study of different types of stability of random/stochastic functional systems, specifically, stochastic delay differential equations (SDDEs). Introducing appropriate Lyapunov functionals enables us to investigate the necessary conditions for stochastic stability, asymptotic stochastic stability, asymptotic mean square stability, mean square exponential stability, global exponential mean square stability, and practical uniform exponential stability. Some examples with numerical simulations are presented to strengthen the theoretical results. Using our theoretical study, important aspects of epidemiological and ecological mathematical models can be revealed. In ecology, the dynamics of Nicholson’s blowflies equation is studied. Conditions of stochastic stability and stochastic global exponential stability of the equilibrium point at which the blowflies become extinct are investigated. In finance, the dynamics of the Black–Scholes market model driven by a Brownian motion with random variable coefficients and time delay is also studied.

Application of hybrid asymptotic methods and modern software for mathematical models developing for nonlinear dynamics of structures with variables in time parameters

The relevance. The aerospace domain requires studies of mathematical models of nonlinear dynamic structures with time-varying parameters.
 The aim of the work. To obtain an approximate analytical solution of nonlinear forced oscillations of the designed models with time-dependent parameters.
 The research methods. A hybrid approach based on perturbation methods, phase integrals, Galorkin orthogonalization criterion is used to obtain solutions.
 Results. Nonlocal investigation of nonlinear systems behavior is done using results of analytical and numerical methods and developed software. Despite the existence of sufficiently powerful numerical software systems, qualitative analysis of nonlinear systems with variable parameters requires improved mathematical models based on effective analytical, including approximate, solutions, which using numerical methods allow to provide a reliable analysis of the studied structures at the stage designing. An approximate solution in analytical form is obtained with constant coefficients that depend on the initial conditions.
 Conclusions. The approximate analytical results and direct numerical solutions of the basic equation were compared which showed a sufficient correlation of the obtained analytical solution. The proposed algorithm and program for visualization of a nonlinear dynamic process could be implemented in nonlinear dynamics problems of systems with time-dependent parameters.