Target layer state estimation in multi-layer complex dynamical networks considering nonlinear node dynamics

In many engineering networks, only a part of target state variables are required to be estimated.On the other hand,multi-layer complex network exists widely in practical situations.In this paper, the state estimation of target state variables in multi-layer complex dynamical networks with nonlinear node dynamics is studied.A suitable functional state observer is constructed with the limited measurement.The parameters of the designed functional observer are obtained from the algebraic method and the stability of the functional observer is proven by the Lyapunov theorem.Some necessary conditions that need to be satisfied for the design of the functional state observer are obtained.Different from previous studies, in the multi-layer complex dynamical network with nonlinear node dynamics, the proposed method can estimate the state of target variables on some layers directly instead of estimating all the individual states.Thus, it can greatly reduce the placement of observers and computational cost.Numerical simulations with the three-layer complex dynamical network composed of three-dimensional nonlinear dynamical nodes are developed to verify the effectiveness of the method.

Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type

We propose a reverse-space nonlocal nonlinear self-dual network equation under special symmetry reduction,which may have potential applications in electric circuits.Nonlocal infinitely many conservation laws are constructed based on its Lax pair.Nonlocal discrete generalized(m,N−m)-fold Darboux transformation is extended and applied to solve this system.As an application of the method,we obtain multi-soliton solutions in zero seed background via the nonlocal discrete N-fold Darboux transformation and rational solutions from nonzero-seed background via the nonlocal discrete generalized(1,N−1)-fold Darboux transformation,respectively.By using the asymptotic and graphic analysis,structures of one-,two-,three-and four-soliton solutions are shown and discussed graphically.We find that single component field in this nonlocal system displays unstable soliton structure whereas the combined potential terms exhibit stable soliton structures.It is shown that the soliton structures are quite different between discrete local and nonlocal systems.Results given in this paper may be helpful for understanding the electrical signals propagation.

DESIGN OF NONLINEAR OBSERVER FOR NONLINEAR SYSTEM BASED ON RBF NEURAL NETWORKS

A new type of nonlinear observer for nonlinear systems is presented. Instead of approximating thc cntire nonlinear system with the neural network （NN）, only the un-modeled part left over after the lincarization is approximated. Compared with the conventional linear observer, the observer provides more accurate estimation of the state. The state estimation error is proved to asymptotically approach zero with the Lyapunov method. The simulation result shows that the proposed observer scheme is effective and has a potential application ability in the fault detection and identification （FDI）, and the state estimation.

Distributed Containment Control of Networked Nonlinear Second-order Systems With Unknown Parameters

In this paper, we study the containment control problem for nonlinear second-order systems with unknown parameters and multiple stationary/dynamic leaders. The topologies that characterize the interaction among the leaders and the followers are directed graphs. Necessary and sufficient criteria which guarantee the control objectives are established for both stationary leaders(regulation case) and dynamic leaders(dynamic tracking case) based protocols. The final states of all the followers are exclusively determined by the initial values of the leaders and the topology structures. In the regulation case, all the followers converge into the convex hull spanned by the leaders,while in the dynamic tracking case, not only the positions of the followers converge into the convex hull but also the velocities of the followers converge into the velocity convex hull of the leaders.Finally, all the theoretical results are illustrated by numerical simulations.

Unsupervised neural network model optimized with evolutionary computations for solving variants of nonlinear MHD Jeffery-Hamel problem

A heuristic technique is developed for a nonlinear magnetohydrodynamics （MHD） Jeffery-Hamel problem with the help of the feed-forward artificial neural net- work （ANN） optimized with the genetic algorithm （GA） and the sequential quadratic programming （SQP） method. The twodimensional （2D） MHD Jeffery-Hamel problem is transformed into a higher order boundary value problem （BVP） of ordinary differential equations （ODEs）. The mathematical model of the transformed BVP is formulated with the ANN in an unsupervised manner. The training of the weights of the ANN is carried out with the evolutionary calculation based on the GA hybridized with the SQP method for the rapid local convergence. The proposed scheme is evaluated on the variants of the Jeffery-Hamel flow by varying the Reynold number, the Hartmann number, and the an- gles of the walls. A large number of simulations are performed with an extensive analysis to validate the accuracy, convergence, and effectiveness of the scheme. The comparison of the standard numerical solution and the analytic solution establishes the correctness of the proposed designed methodologies.

Visual analytics tool for the interpretation of hidden states in recurrent neural networks

In this paper,we introduce a visual analytics approach aimed at helping machine learning experts analyze the hidden states of layers in recurrent neural networks.Our technique allows the user to interactively inspect how hidden states store and process information throughout the feeding of an input sequence into the network.The technique can help answer questions,such as which parts of the input data have a higher impact on the prediction and how the model correlates each hidden state configuration with a certain output.Our visual analytics approach comprises several components:First,our input visualization shows the input sequence and how it relates to the output(using color coding).In addition,hidden states are visualized through a nonlinear projection into a 2-D visualization space using t-distributed stochastic neighbor embedding to understand the shape of the space of the hidden states.Trajectories are also employed to show the details of the evolution of the hidden state configurations.Finally,a time-multi-class heatmap matrix visualizes the evolution of the expected predictions for multi-class classifiers,and a histogram indicates the distances between the hidden states within the original space.The different visualizations are shown simultaneously in multiple views and support brushing-and-linking to facilitate the analysis of the classifications and debugging for misclassified input sequences.To demonstrate the capability of our approach,we discuss two typical use cases for long short-term memory models applied to two widely used natural language processing datasets.

A Prediction Method Based on Improved Echo State Network for COVID-19 Nonlinear Time Series

This paper proposes a prediction method based on improved Echo State Network for COVID-19 nonlinear time series, which improves the Echo State Network from the reservoir topology and the output weight matrix, and adopt the ABC (Artificial Bee Colony) algorithm based on crossover and crowding strategy to optimize the parameters. Finally, the proposed method is simulated and the results show that it has stronger prediction ability for COVID-19 nonlinear time series.

Fault detection for nonlinear networked control systems based on fuzzy observer

Security and reliability must be focused on control sys- tems firstly, and fault detection and diagnosis （FDD） is the main theory and technology. Now, there are many positive results in FDD for linear networked control systems （LNCSs）, but nonlinear networked control systems （NNCSs） are less involved. Based on the T-S fuzzy-modeling theory, NNCSs are modeled and network random time-delays are changed into the unknown bounded uncertain part without changing its structure. Then a fuzzy state observer is designed and an observer-based fault detection approach for an NNCS is presented. The main results are given and the relative theories are proved in detail. Finally, some simulation results are given and demonstrate the proposed method is effective.

A New Method for Solving Nonlinear Partial Differential Equations Based on Liquid Time-Constant Networks

In this paper,physics-informed liquid networks(PILNs)are proposed based on liquid time-constant networks(LTC)for solving nonlinear partial differential equations(PDEs).In this approach,the network state is controlled via ordinary differential equations(ODEs).The significant advantage is that neurons controlled by ODEs are more expressive compared to simple activation functions.In addition,the PILNs use difference schemes instead of automatic differentiation to construct the residuals of PDEs,which avoid information loss in the neighborhood of sampling points.As this method draws on both the traveling wave method and physics-informed neural networks(PINNs),it has a better physical interpretation.Finally,the KdV equation and the nonlinear Schr¨odinger equation are solved to test the generalization ability of the PILNs.To the best of the authors’knowledge,this is the first deep learning method that uses ODEs to simulate the numerical solutions of PDEs.

Adaptive Backstepping Output Feedback Control for SISO Nonlinear System Using Fuzzy Neural Networks

In this paper, a new fuzzy-neural adaptive control approach is developed for a class of single-input and single-output （SISO） nonlinear systems with unmeasured states. Using fuzzy neural networks to approximate the unknown nonlinear functions, a fuzzy- neural adaptive observer is introduced for state estimation as well as system identification. Under the framework of the backstepping design, fuzzy-neural adaptive output feedback control is constructed recursively. It is proven that the proposed fuzzy adaptive control approach guarantees the global boundedness property for all the signals, driving the tracking error to a small neighbordhood of the origin. Simulation example is included to illustrate the effectiveness of the proposed approach.

Optimal State Estimation and Fault Diagnosis for a Class of Nonlinear Systems

This study proposes a scheme for state estimation and,consequently,fault diagnosis in nonlinear systems.Initially,an optimal nonlinear observer is designed for nonlinear systems subject to an actuator or plant fault.By utilizing Lyapunov's direct method,the observer is proved to be optimal with respect to a performance function,including the magnitude of the observer gain and the convergence time.The observer gain is obtained by using approximation of Hamilton-Jacobi-Bellman(HJB)equation.The approximation is determined via an online trained neural network(NN).Next a class of affine nonlinear systems is considered which is subject to unknown disturbances in addition to fault signals.In this case,for each fault the original system is transformed to a new form in which the proposed optimal observer can be applied for state estimation and fault detection and isolation(FDI).Simulation results of a singlelink flexible joint robot(SLFJR)electric drive system show the effectiveness of the proposed methodology.

Turbo-shaft engine adaptive neural network control based on nonlinear state space equation

Intelligent Adaptive Control(AC) has remarkable advantages in the control system design of aero-engine which has strong nonlinearity and uncertainty. Inspired by the Nonlinear Autoregressive Moving Average(NARMA)-L2 adaptive control, a novel Nonlinear State Space Equation(NSSE) based Adaptive neural network Control(NSSE-AC) method is proposed for the turbo-shaft engine control system design. The proposed NSSE model is derived from a special neural network with an extra layer, and the rotor speed of the gas turbine is taken as the main state variable which makes the NSSE model be able to capture the system dynamic better than the NARMA-L2 model. A hybrid Recursive Least-Square and Levenberg-Marquardt(RLS-LM) algorithm is advanced to perform the online learning of the neural network, which further enhances both the accuracy of the NSSE model and the performance of the adaptive controller. The feedback correction is also utilized in the NSSE-AC system to eliminate the steady-state tracking error. Simulation results show that, compared with the NARMA-L2 model, the NSSE model of the turboshaft engine is more accurate. The maximum modeling error is decreased from 5.92% to 0.97%when the LM algorithm is introduced to optimize the neural network parameters. The NSSE-AC method can not only achieve a better main control loop performance than the traditional controller but also limit all the constraint parameters efficiently with quick and accurate switching responses even if component degradation exists. Thus, the effectiveness of the NSSE-AC method is validated.

An Embedded Consensus ADMM Distribution Algorithm Based on Outer Approximation for Improved Robust State Estimation of Networked Microgrids

Networked microgrids(NMGs)are critical in theaccommodation of distributed renewable energy.However,theexisting centralized state estimation(SE)cannot meet the demandsof NMGs in distributed energy management.The currentestimator is also not robust against bad data.This study introducesthe concepts of relative error to construct an improvedrobust SE(IRSE)optimization model with mixed-integer nonlinearprogramming(MINLP)that overcomes the disadvantage ofinaccurate results derived from different measurements whenthe same tolerance range is considered in the robust SE(RSE).To improve the computation efficiency of the IRSE optimizationmodel,the number of binary variables is reduced based on theprojection statistics and normalized residual methods,which effectivelyavoid the problem of slow convergence or divergenceof the algorithm caused by too many integer variables.Finally,an embedded consensus alternating direction of multiplier method(ADMM)distribution algorithm based on outer approximation(OA)is proposed to solve the IRSE optimization model.This algorithm can accurately detect bad data and obtain SE resultsthat communicate only the boundary coupling informationwith neighbors.Numerical tests show that the proposed algorithmeffectively detects bad data,obtains more accurate SE results,and ensures the protection of private information in all microgrids.

H∞-Feedback Control of Heparin-Controlled Blood Clotting Network for Cardiac Surgeries

This paper presents a solution methodology for H∞-feedback control design problem of Heparin controlled blood clotting network under the presence of stochastic noise. The formulaic solution procedure to solve nonlinear partial differential equation, the Hamilton-Jacobi-Isaacs equation with Successive Galrkin’s Approximation is sketched and validity is proved. According to Lyapunov’s theory, with solutions of the nonlinear PDEs, robust feedback control is designed. To confirm the performance and robustness of the designed controller, numerical and Monte-Carlo simulation results by Simulink software on MATLAB are provided.

SYNCHRONIZATION OF COMPLEX NETWORKS VIA A SIMPLE AND ECONOMICAL FIXED-TIME CONTROLLER

1 Introduction and Main Results Consider the following differential equation x(t)=f(x(t)),x(0)=x0,where x∈Rn denotes the state variable of system(1.1),f:Rn→Rn is a nonlinear vector field and x0 is the initial value of the system.

H2 and H∞-Feedback Control Design for Nonlinear Gene Networks via Successive Galerkin’s Approximation

This paper presents a design method of H2 and H∞-feedback control loop for nonlinear smooth gene networks that are in control affine form. Formulaic solution methodology for solving the nonlinear partial differential equations, namely the Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations through successive Galerkin’s approximation is implemented and the results are compared. Throughout the implementation, there were several caveats that need to be further resolved for practical applications in general cases. Such issues and the clarification of causes are mathematically established and reviewed.

From masses and radii of neutron stars to EOS of nuclear matter through neural network

The equation of state(EOS)of dense nuclear matter is a key factor for determining the internal structure and properties of neutron stars.However,the EOS of high-density nuclear matter has great uncertainty,mainly because terrestrial nuclear experiments cannot reproduce matter as dense as that in the inner core of a neutron star.Fortunately,continuous improvements in astronomical observations of neutron stars provide the opportunity to inversely constrain the EOS of high-density nuclear matter.Several methods have been proposed to implement this inverse constraint,including the Bayesian analysis algorithm,the Lindblom’s approach,and so on.Neural network algorithm is an effective method developed in recent years.By employing a set of isospin-dependent parametric EOSs as the training sample of a neural network algorithm,we set up an effective way to reconstruct the EOS with relative accuracy using a few mass-radius data.Based on the obtained neural network algorithms and according to the NICER observations on masses and radii of neutron stars with assumed precision,we obtain the inversely constrained EOS and further calculate the corresponding macroscopic properties of the neutron star.The results are basically consistent with the constraint on EOS in Huth et al.[Nature 606,276(2022)]based on Bayesian analysis.Moreover,the results show that even though the neural network algorithm was obtained using the finite parameterized EOS as the training set,it is valid for any rational parameter combination of the parameterized EOS model.

Novel Multisoliton Solutions of Some Soliton Equations

The novel multisoliton solutions for the nonlinear lumped self-dual network equations, Toda lattice and KP equation were obtained by using the Hirota direct method.

Study on the prediction and inverse prediction of detonation properties based on deep learning

The accurate and efficient prediction of explosive detonation properties has important engineering significance for weapon design.Traditional methods for predicting detonation performance include empirical formulas,equations of state,and quantum chemical calculation methods.In recent years,with the development of computer performance and deep learning methods,researchers have begun to apply deep learning methods to the prediction of explosive detonation performance.The deep learning method has the advantage of simple and rapid prediction of explosive detonation properties.However,some problems remain in the study of detonation properties based on deep learning.For example,there are few studies on the prediction of mixed explosives,on the prediction of the parameters of the equation of state of explosives,and on the application of explosive properties to predict the formulation of explosives.Based on an artificial neural network model and a one-dimensional convolutional neural network model,three improved deep learning models were established in this work with the aim of solving these problems.The training data for these models,called the detonation parameters prediction model,JWL equation of state(EOS)prediction model,and inverse prediction model,was obtained through the KHT thermochemical code.After training,the model was tested for overfitting using the validation-set test.Through the model-accuracy test,the prediction accuracy of the model for real explosive formulations was tested by comparing the predicted value with the reference value.The results show that the model errors were within 10%and 3%for the prediction of detonation pressure and detonation velocity,respectively.The accuracy refers to the prediction of tested explosive formulations which consist of TNT,RDX and HMX.For the prediction of the equation of state for explosives,the correlation coefficient between the prediction and the reference curves was above 0.99.For the prediction of the inverse prediction model,the prediction error of the explosive equation was within 9%.This indicates that the models have utility in engineering.

Partial Topology Identification of Stochastic Multi-Weighted Complex Networks Based on Graph-Theoretic Method and Adaptive Synchronization

This article aims to identify the partial topological structures of delayed complex network.Based on the drive-response concept,a more universal model,which includes nonlinear couplings,stochastic perturbations and multi-weights,is considered into drive-response networks.Different from previous methods,we obtain identification criteria by combining graph-theoretic method and adaptive synchronization.After that,the partial topological structures of stochastic multi-weighted complex networks with or without time delays can be identified successfully.Moreover,response network can reach synchronization with drive network.Ultimately,the effectiveness of the proposed theoretical results is validated through numerical simulations.