Class Of Nonlinear Dynamical Systems Research Articles

Dark Type Dynamical Systems: The Integrability Algorithm and Applications

Based on a devised gradient-holonomic integrability testing algorithm, we analyze a class of dark type nonlinear dynamical systems on spatially one-dimensional functional manifolds possessing hidden symmetry properties and allowing their linearization on the associated cotangent spaces. We described main spectral properties of nonlinear Lax type integrable dynamical systems on periodic functional manifolds particular within the classical Floquet theory, as well as we presented the determining functional relationships between the conserved quantities and related geometric Poisson and recursion structures on functional manifolds. For evolution flows on functional manifolds, parametrically depending on additional functional variables, naturally related with the classical Bellman-Pontriagin optimal control problem theory, we studied a wide class of nonlinear dynamical systems of dark type on spatially one-dimensional functional manifolds, which are both of diffusion and dispersion classes and can have interesting applications in modern physics, optics, mechanics, hydrodynamics and biology sciences. We prove that all of these dynamical systems possess rich hidden symmetry properties, are Lax type linearizable and possess finite or infinite hierarchies of suitably ordered conserved quantities.

THE SYNTHESIS OF MATHEMATICAL MODELS OF NONLINEAR DYNAMIC SYSTEMS USING VOLTERRA INTEGRAL EQUATION

The problem of creating mathematical models of nonlinear dynamical systems does not have an unambiguous solution and requires the creation of a separate synthesis method for each such object. To develop a method for synthesizing mathematical models of an extensive class of nonlinear dynamical systems with polynomial nonlinearities. The work uses a method based on the solution of the Volterra integral equation in the ideology set forth in Van Trees H.L., according to which the structure of a nonlinear dynamical object present47s a series connection of the linear part, characterizing the inertial properties of the system, and the nonlinear element, given by static characteristic. The difference of the suggested version of the method from the classical one, proposed in the works of Van Trees H.L., is an expansion of their input and output signals into Fourier series and a representation of the inertial part of these systems by their Bode plots, connected into one structure with input and output signals and non-linearity by Volterra integral equation. The algorithm of the proposed method is disclosed by the example of solving the problem of identifying a nonlinear dynamical system which impulse response of the inertial part satisfies the separability requirement, the order of the polynomial nonlinearity is three, and the model of the input signal has the form of a sinusoid "raised" over the time axis on a priori given constant level. A computational experiment was carried out on the example of nonlinear dynamical systems with the third order of the nonlinear characteristic and the first and second orders of the model of the inertial part of these systems with the specified algorithms of their parametric identification. The suggested method allows to synthesis the mathematical model of a nonlinear dynamical system with the polynomial static characteristic to the case when the input signal has an arbitrary number of harmonics, and the model of the inertial part and the nonlinear polynomial function have an arbitrary order.

Universal constraint on nonlinear population dynamics

Ecological and evolutionary processes show various population dynamics depending on internal interactions and environmental changes. While crucial in predicting biological processes, discovering general relations for such nonlinear dynamics has remained a challenge. Here, we derive a universal information-theoretical constraint on a broad class of nonlinear dynamical systems represented as population dynamics. The constraint is interpreted as a generalization of Fisher’s fundamental theorem of natural selection. Furthermore, the constraint indicates nontrivial bounds for the speed of critical relaxation around bifurcation points, which we argue are universally determined only by the type of bifurcation. Our theory is verified for an evolutionary model and an epidemiological model, which exhibit the transcritical bifurcation, as well as for an ecological model, which undergoes limit-cycle oscillation. This work paves a way to predict biological dynamics in light of information theory, by providing fundamental relations in nonequilibrium statistical mechanics of nonlinear systems.

Sampled-data extremum-seeking framework for constrained optimization of nonlinear dynamical systems

Most extremum-seeking control (ESC) approaches focus solely on the problem of finding the extremum of some unknown, steady-state input–output map, providing parameter settings that lead to optimal steady-state system performance. However, many industrial applications also have to deal with constraints on operating conditions due to, e.g., actuator limitations, limitations on tunable system parameters, or constraints on measurable variables. In particular, constraints on measurable variables are typically unknown in terms of their relationship with the tunable system parameters. In addition, the constraints on system inputs as a result of the constraints on measurable variables may conflict with the otherwise optimal operational condition, and hence should be taken into account in the data-based optimization approach. In this work, we propose a sampled-data extremum-seeking framework for the constrained optimization of a class of nonlinear dynamical systems with measurable constrained variables. In this framework, barrier function methods are employed, exploiting both the objective function and constraint functions which are available through output measurement only. We show, under the assumption that the parametric initialization yield operating conditions that do not violate the constraints, that (1) the resulting closed-loop dynamics is stable, (2) constraint satisfaction of the inputs is guaranteed for all iterations of the optimization process, and (3) constrained optimization is achieved. We illustrate the working principle of the proposed framework by means of an industrial case study of the constrained optimization of extreme ultraviolet light generation in a laser-produced plasma source within a state-of-the-art lithography system.

Predefined-time synchronization of 5D Hindmarsh–Rose neuron networks via backstepping design and application in secure communication

In this paper, the fast synchronization problem of 5D Hindmarsh–Rose neuron networks is studied. Firstly, the global predefined-time stability of a class of nonlinear dynamical systems is investigated under the complete beta function. Then an active controller via backstepping design is proposed to achieve predefined-time synchronization of two 5D Hindmarsh–Rose neuron networks in which the synchronization time of each state variable of the master-slave 5D Hindmarsh–Rose neuron networks is different and can be defined in advance, respectively. To show the applicability of the obtained theoretical results, the designed predefined-time backstepping controller is applied to secure communication to realize asynchronous communication of multiple different messages. Three numerical simulations are provided to validate the theoretical results.

A mode-in-state contribution factor based on Koopman operator and its application to power system analysis

This paper proposes a mode-in-state contribution factor for a class of nonlinear dynamical systems by utilizing spectral properties of the Koopman operator and sensitivity analysis. Using eigenfunctions of the Koopman operator for a target nonlinear system, we show that the relative contribution between modes and state variables can be quantified beyond a linear regime, where the nonlinearity of the system is taken into consideration. The proposed contribution factor is applied to the numerical analysis of large-signal simulations for an interconnected AC/multi-terminal DC power system.

A Convex Parameterization of Robust Recurrent Neural Networks

Recurrent neural networks (RNNs) are a class of nonlinear dynamical systems often used to model sequence-to-sequence maps. RNNs have excellent expressive power but lack the stability or robustness guarantees that are necessary for many applications. In this letter, we formulate convex sets of RNNs with stability and robustness guarantees. The guarantees are derived using incremental quadratic constraints and can ensure global exponential stability of all solutions, and bounds on incremental ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gain (the Lipschitz constant of the learned sequence-to-sequence mapping). Using an implicit model structure, we construct a parametrization of RNNs that is jointly convex in the model parameters and stability certificate. We prove that this model structure includes all previously-proposed convex sets of stable RNNs as special cases, and also includes all stable linear dynamical systems. Numerical experiments illustrate the utility of the proposed model class in the context of non-linear system identification.

Estimation of attraction domain of nonlinear systems with saturated impulses and input disturbances

AbstractIn this article, the estimation of attraction domain of nonlinear systems with saturated impulses and input disturbances is investigated. The sufficient conditions which can ensure the robust convergence of a class of nonlinear dynamical systems for every initial condition in the attraction domain are established. Moreover, some optimization problems are proposed to design the control gain so as to enlarge the size of the estimation of attraction domain as much as possible with a prespecified decay rate. In addition, the exponential stabilization and the estimation of attraction domain for nonlinear dynamical systems under this saturated impulsive control without disturbances are also studied. Finally, simulation results for a chaotic oscillator are displayed to show the effectiveness and feasibility of our results.

Analysis of Learning Algorithms for Multilayer Neural Networks

The modern stage of development of science and technology is characterized by a rapid increase in the complexity of the created technical systems. The management of such systems requires the development of new management methods, since the modification and improvement of traditional management techniques does not always ensure the fulfillment of stringent requirements for management quality indicators. Classical control methods are mainly based on the theory of linear systems, while most real objects are non-linear. The problem of the synthesis of control systems under conditions of uncertainty is currently one of the central problems in the modern theory of automatic control. The complexity of the control object itself, structural, parametric and information uncertainties in the description of the control object, and the complexity of control problems, the multi criteria of optimization problems, the lack of possible analytical solutions, the need to take into account all the properties of disturbances, etc. The solution to this problem requires a search for alternative approaches to the design of control systems, one of which involves the introduction of neural network systems. Neural network control systems are a high-tech direction of control theory and belong to the class of nonlinear dynamic systems. High performance due to parallelization of input information in combination with the ability to train neural networks makes this technology very attractive for creating control devices in automatic systems. Neural networks can be used to build regulating and switching devices, reference, adaptive, nominal and inverse-dynamic models of objects, on the basis of which objects are studied, analysis of the influence of disturbances acting on an object, determination of the optimal control law, search or calculating the optimal program for changing the impact when changing the values of the parameters of the object and the characteristics of the input data. In addition, neural networks can be used to identify objects, predict the state of objects, recognize, cluster, classify, analyze a large amount of data arriving at high speed from a large number of devices and sensors, and the like. The ability to learn according to a given principle of functioning allows creating automated control systems that are optimal in terms of speed, energy consumption, etc. Naturally, in this case, it is possible to implement several principles of functioning and the transition from one to another. They are a universal tool for modeling multidimensional nonlinear objects and finding solutions to ill-posed problems.

Nonlinear Network Dynamics with Consensus\u2013Dissensus Bifurcation

We study a nonlinear dynamical system on networks inspired by the pitchfork bifurcation normal form. The system has several interesting interpretations: as an interconnection of several pitchfork systems, a gradient dynamical system and the dominating behaviour of a general class of nonlinear dynamical systems. The equilibrium behaviour of the system exhibits a global bifurcation with respect to the system parameter, with a transition from a single constant stationary state to a large range of possible stationary states. Our main result classifies the stability of (a subset of) these stationary states in terms of the effective resistances of the underlying graph; this classification clearly discerns the influence of the specific topology in which the local pitchfork systems are interconnected. We further describe exact solutions for graphs with external equitable partitions and characterize the basins of attraction on tree graphs. Our technical analysis is supplemented by a study of the system on a number of prototypical networks: tree graphs, complete graphs and barbell graphs. We describe a number of qualitative properties of the dynamics on these networks, with promising modelling consequences.

On Analytical Construction of Observable Functions in Extended Dynamic Mode Decomposition for Nonlinear Estimation and Prediction

We propose an analytical construction of observable functions in the extended dynamic mode decomposition (EDMD) algorithm. EDMD is a numerical method for approximating the spectral properties of the Koopman operator. The choice of observable functions is fundamental for the application of EDMD to nonlinear problems arising in systems and control. Existing methods either start from a set of dictionary functions and look for the subset that best fits the underlying nonlinear dynamics or they rely on machine learning algorithms to “learn” observable functions. Conversely, in this letter, we start from the dynamical system model and lift it through the Lie derivatives, rendering it into a polynomial form. This proposed transformation into a polynomial form is exact, and it provides an adequate set of observable functions. The strength of the proposed approach is its applicability to a broader class of nonlinear dynamical systems, particularly those with nonpolynomial functions and compositions thereof. Moreover, it retains the physical interpretability of the underlying dynamical system and can be readily integrated into existing numerical libraries. The proposed approach is illustrated with an application to electric power systems. The modeled system consists of a single generator connected to an infinite bus, where nonlinear terms include sine and cosine functions. The results demonstrate the effectiveness of the proposed procedure in off-attractor nonlinear dynamics for estimation and prediction; the observable functions obtained from the proposed construction outperform methods that use dictionary functions comprising monomials or radial basis functions.

A Robust Bayesian Approach for Online Filtering in the Presence of Contaminated Observations

This article proposes an online scheme for state estimation of a generic class of nonlinear dynamical systems in the presence of abnormal measurement data from sensors. We thoroughly illustrate why the performance of standard recursive Bayesian inference degrades in the presence of any measurement distortion. After demonstrating how different abnormalities can be accommodated explicitly inside a generic state-space model, we propose a robust mechanism to perform recursive Bayesian inference on the presented model to not only detect but also mitigate the effect of corrupted measurements in the final state estimates. Using simulations and experimentation, we demonstrate the success of the proposed framework in reducing the impact of different types of distortions in measurements. The ability to tackle different kinds of measurement abnormalities during online inference sets the proposed method apart from the existing techniques.

A Continuous-Discrete Finite Memory Observer Design for a Class of Nonlinear Systems: Application to Fault Diagnosis

This paper aims to develop a continuous-discrete finite memory observer (CD-FMO) for a class of nonlinear dynamical systems modeled by ordinary differential equations (ODEs) with discrete measurements. The nonlinear systems under consideration are at least locally Lipschitz, which guarantees the existence and uniqueness of solution at each time instant. The proposed nonlinear observer uses a finite number of collected measurements to estimate the system state in the presence of measurement noise. Besides, a one-step prediction algorithm incorporated with an iterative-update scheme is performed to solve the integral problem caused by system nonlinearity, and an analysis of the numerical integration approximation error is given. The properties of estimation performance have been further proved in deterministic case and been analyzed by Monte Carlo simulation in stochastic cases. It is worth noting that the presented method has a finite-time convergence, while most nonlinear observers are usually asymptotically convergent. Another advantage of CD-FMO is that it has no initial value problem. For the application purpose, residuals are generated to implement fault detection cooperated with Cumulative Sum (CUSUM) control charts, while a bank of CD-FMOs is adopted to realize fault isolation for different sensor and actuator faults of the considered nonlinear robotic arm. The robustness and effectiveness of the proposed approach are illustrated via the simulation results.

A Novel TRUST-TECH-Enabled Trajectory-Unified Methodology for Computing Multiple Optimal Solutions of Constrained Nonlinear Optimization: Theory and Computation

A novel trajectory-unified (TJU) methodology, enabled by transformation under stability-retraining equilibria characterization (TRUST-TECH, TT) method, is proposed to systematically and deterministically compute a set of multiple local optimal solutions (LOSs), if not all of them, for constrained nonlinear optimization problems. The proposed TT-TJU methodology is composed of two key phases. During Phase I, it robustly computes a LOS; while during Phase II, it systematically computes a set of neighboring LOSs. A theoretical foundation is developed for the proposed TT-TJU methodology. The existence of an optimal solution is shown to be equivalent to the existence of a stable equilibrium point of a class of nonlinear dynamical system derived from the constrained optimization problem under study. From a computational viewpoint, Phase I is numerically implemented in a 3-stage procedure for fast and robust computation of an optimal solution. The second phase makes use of the TRUST-TECH methodology to lead the search away from a computed optimal solution and continue to deterministically compute neighboring LOSs. The proposed methodology has been evaluated on an illustrative bivariate example and a benchmark problem with 110 variables and 55 constraints with promising results.

Observer normal forms for a class of nonlinear systems by means of coupled auxiliary dynamics

SummaryThis work aims at developing a class of nonlinear dynamical systems that can be converted into nonlinear observer normal forms, which support the well‐known high gain observer. This will be performed by means of the so‐called dynamics extension method, which explicitly supplies a set of auxiliary dynamics and changes of coordinates. In this work, the target nonlinear observer normal forms depend on both the output of the studied dynamical systems and the auxiliary variables involved in the definition of the extended dynamics. The method is proposed in a comprehensible way because it does not use differential geometry techniques, such as the Lie brackets calculations of some vector fields. The proposed class of dynamical systems has many practical applications such as epidemic systems, social networks, and so on. To highlight the effectiveness of this algorithm, it supplies an example based on the Maternal Immune Susceptible Exposed Infected and Recovered model.

Sparse RKHS estimation via globally convex optimization and its application in LPV-IO identification

Function estimation using the Reproducing Kernel Hilbert Space (RKHS) framework is a powerful tool for identification of a general class of nonlinear dynamical systems without requiring much a priori information on model orders and nonlinearities involved. However, the high degrees-of-freedom (DOFs) of RKHS estimators has its price, as in case of large scale function estimation problems, they often require a serious amount of data samples to explore the search space adequately for providing high-performance model estimates. In cases where nonlinear dynamic relations can be expressed as a sum of functions, the literature proposes solutions to this issue by enforcing sparsity for adequate restriction of the DOFs of the estimator, resulting in parsimonious model estimates. Unfortunately, all existing solutions are based on greedy approaches, leading to optimization schemes which cannot guarantee convergence to the global optimum. In this paper, we propose an ℓ1-regularized non-parametric RKHS estimator which is the solution of a quadratic optimization problem. Effectiveness of the scheme is demonstrated on the non-parametric identification problem of LPV-IO models where the method solves simultaneously (i) the model order selection problem (in terms of number of input–output lags and input delay in the model structure) and (ii) determining the unknown functional dependency of the model coefficients on the scheduling variable directly from data. The paper also provides an extensive simulation study to illustrate effectiveness of the proposed scheme.

Periodic behavior of a class of nonlinear dynamic systems based on the Runge-Kutta algorithm

Periodic behavior of a class of nonlinear dynamic systems based on the Runge-Kutta algorithm

A hybrid impulsive and sampled-data control framework for a class of nonlinear dynamical systems with input constraints

In this article, a new hybrid controller is designed, which consists of sampled-data controller and impulsive controller with actuator saturation. Firstly, by applying polytopic representation approach and establishing an appropriated Lyapunov function, some established linear matrix inequalities (LMIs) are obtained to guarantee exponential stabilization of a class of nonlinear dynamical systems. It is worth mentioning that these LMIs are very easy to verify by the Matlab LMI toolbox. Moreover, based on sector nonlinearity model approach, sufficient conditions ensuring exponential stabilization are established by LMIs. Furthermore, it is also shown that a hybrid controller without actuator saturation and under same assumptions can achieve exponential stabilization of a class of nonlinear dynamical systems. Finally, three examples are provided to show the effectiveness of theoretical analysis.

Adaptive smooth control for nonlinear uncertain systems

This paper presents an adaptive smooth controller for a class of nonlinear dynamical systems in the presence of bounded uncertainties with unknown bounds. Motivated by the concept of sliding mode control, a continuous control law that drives the sliding variable to a user-specified small domain in a finite time is developed. Also, an adaptation law is obtained to ensure that the gain update is performed in real time and the error is always bounded within a desired range without a priori knowledge of the uncertainties. Only the magnitude of the control input is required for online gain adaptation, and hence, the gain update is readily implemented in real time, which greatly eases the application of the suggested algorithm to real-world systems. In addition, by adding nonlinear terms in the sliding manifold, the reaching phase vanishes and the transient performance of the controlled system is considerably improved. Numerical examples serve to demonstrate the effectiveness and the robustness of the control methodology proposed herein.

Study on chaotic response of a class of nonlinear dynamical system by path integral method

Path integration; Probability density; Nonlinear stochastic dynamical systems