Complex Nonlinear Systems Research Articles

Neural Network Design and Training for Longitudinal Flight Control of a Tilt-Rotor Hybrid Vertical Takeoff and Landing Unmanned Aerial Vehicle

This paper considers a hybrid vertical take-off and landing (VTOL) unmanned aerial vehicle (UAV). By tilting its propellers, the aircraft can transition from rotary-wing (RW) multirotor mode to fixed-wing (FW) mode and vice versa. A novel architecture of a neural network-based controller (NNC) is presented. An “imitative learning” approach is employed to train the NNC to mimic the response of an expert but computationally expensive model predictive controller (MPC). The resulting NNC approximates the MPC’s solution while significantly decreasing the computational cost. The NNC is trained on the longitudinal axis. Successful simulations and real flight tests prove that the NNC is suitable for the longitudinal axis control of a complex nonlinear system such as the tilt-rotor VTOL UAV through a sequence of transitions between the RW mode to the FW mode, and vice versa, in a forward flight.

Fractional View Analysis of Coupled Whitham-Broer-Kaup Equations Arising in Shallow Water with Caputo Derivative

This research explores the analysis of the nonlinear fractional systems described by the Whitham-Broer-Kaup equations using novel mathematical tools like the Aboodh transform iteration method and the Aboodh residual power series method in light of the Caputo operator theory. The Whitham-Broer-Kaup equations are critical for describing the nonlinear propagation of dispersive waves and have enormous practical relevance. In the application of the Aboodh transform iteration method and the Aboodh residual power series method, as well as the introduction of the Caputo operator, the modelling accuracy is improved by calculating the fractional derivatives. The study has proven the usefulness of these techniques in providing solutions to fractional nonlinear systems that are only approximate but are insightful concerning dynamic behaviour. The insertion of the Caputo operator in the modelling method gives it some sophistication, simulating the non-locality inherent in fractional calculus. Consequently, this study enriches mathematical models and computational approaches, bringing in solid tools for researchers who want to examine complex nonlinear systems with fractional nature.

Tracking control strategy of tendon driven robotic arm under adaptive neural network

IntroductionWith the rapid optimization and evolution of various neural networks, the control problem of robotic arms in the area of automation control has gradually received more attention.MethodsTo improve the control performance of robotic arms under complex dynamic models, this study proposes an adaptive affective radial basis function network control strategy. Firstly, the kinematic and dynamic mathematical models of the tendon driven robotic arm are constructed. Then, by integrating the affective computing model and the radial basis function network, an adaptive affective radial basis function network control algorithm is constructed.Results and DiscussionThe research results indicate that the designed algorithm significantly outperforms the other two compared algorithms in terms of control accuracy and stability. In benchmark performance testing, the designed algorithm has a error accuracy of up to 0.97 and a steady state of up to 0.95. In the simulation results, the maximum torque change of the designed algorithm is only 3.8 Nm, which is much lower than other algorithms. In addition, the control error fluctuation range of this algorithm is between −0.001 and 0.001, almost close to zero error. This study provides a new optimization strategy for precise control of tendon driven robotic arms, and also opens up new avenues for the application of artificial intelligence technology in complex nonlinear system control.

Real-time prediction of structural responses in deep-water jacket platforms using Ada-STLNet: A comprehensive analysis with prototype monitoring data

Real-time prediction of the mechanical behaviors based on prototype monitoring data is crucial for ensuring the integrity and safety of deep-water jacket platforms. As complex nonlinear systems, these platforms’ response exhibit varying temporal trends, dynamic spatial correlations, and load dependencies. This makes accurate prediction of future axial force and bending moment responses a significant challenge. In this paper, a novel deep learning method called Adaptive Spatio-Temporal Load Network (Ada-STLNet) is proposed aiming to address the issue of multi-node axial force and bending moment response prediction of deep-water jacket platform. Our model utilizes an enhanced graph convolution (EGCN) and self-attention mechanism for efficient spatial correlation modeling in multi-node response, LSTM for long sequence temporal feature capture. It integrates spatial and temporal, along with load-aware modules to capture intricate patterns in structural responses. Additionally, a multi-gated coupling mechanism with two independent gating modules is designed to adaptively represent the complex dependencies of structural responses, ensuring model stability and robustness. Extensive experiments conducted on prototype structural monitoring data from a deep-water jacket platform in the South China Sea demonstrate that Ada-STLNet outperforms six traditional models, including HA, SVR, KNN, MLP, LSTM, and GRU, across various prediction steps. The proposed model exhibits superior accuracy, particularly in short-term predictions, achieving up to 62.80% improvement in MAE and 48.11% in RMSE for axial force predictions compared to the second-best model. Ablation studies confirm the effectiveness of each component within Ada-STLNet. This research highlights the potential of Ada-STLNet for reliable and accurate prediction of structural responses, contributing significantly to the field of marine engineering.

Adaptive fixed-time fuzzy control for delayed recycling continuous stirred tank reactor with asymmetric time-varying full-state constraints

Recently, there are some results concerning stability and finite-time stability control for continuous stirred tank reactors (CSTR). However, when considering the control of time-delay CSTR systems or fixed-time stability of CSTR systems, there are relatively few articles involved. In this paper, we first consider the CSTR system that includes both of the above factors simultaneously, that is, the adaptive fixed-time fuzzy control design problem for a CSTR with a delayed recycle stream and asymmetric time-varying full-state constraints. Firstly, based on existing works, we present an improved practical fixed-time stability criterion, which can be applied to more complex nonlinear systems. Then, we construct time-varying asymmetric integral barrier Lyapunov functions to handle asymmetric time-varying full-state constraints. Next, fuzzy logic systems are used to approximate the unknown functions of the system, and a sliding mode differentiator is introduced to avoid the “explosion of complexity” of the time derivative of the virtual controller. Furthermore, using the improved practical fixed-time stability criterion, we present the adaptive fixed-time fuzzy controller design processes that can ensure tracking performance in a fixed-time, and states of the system never violate their constraint bounds. Finally, a simulation example is provided to verify the effectiveness of the proposed control strategy.

High-efficient non-iterative reliability-based design optimization based on the design space virtually conditionalized reliability evaluation method

Dynamic-reliability-based design optimization (DRBDO) is a promising methodology to address the significant challenge posed by the new generation of structural design theories centered around reliability considerations. Solving DRBDO problems typically requires iterations ranging from a dozen to several hundreds, with each iteration dedicated to updating the values of design variables. Furthermore, DRBDO necessitates hundreds of or even more representative structural analyses at each iteration to compute the reliability measure, which serves as a foundation for determining the search direction in the subsequent iteration. This results in the double-loop problem confronted by DRBDO, leading to substantial computational costs for structural re-computing, particularly in the cases involving complex nonlinear stochastic dynamical systems. In the present paper, a non-iterative DRBDO paradigms is proposed by combining a novel virtually conditional reliability evaluation and the newly proposed decoupled multi-probability density evolution method (M-PDEM). By leveraging the decoupled M-PDEM, a series of one-dimensional partial differential equations (PDEs) named Li-Chen equations are solved to calculate the joint PDF of multiple responses. This enables efficient computation of the joint probability density function (PDF) of design variables and extreme response as well as the conditional PDF of the extreme response given the values of design variables based on finite representative structural analyses. Then, the reliability of different designs can be regarded as the integral of the conditional PDF, which yields the reliability feasible domain. For problems that the objective function is monotonic to each design variables, by combining with a direct search technique, this method transforms the optimization process into true iteration-free calculations, and thereby eliminates the significant computational burden associated with structural re-computing at different intermediate designs in optimization iterations. Finally, the accuracy and effectiveness of this novel method are validated through numerical examples.

Modeling and analysis of nonlinear dynamics of axisymmetric vector nozzle based on deep neural network

Abstract The axisymmetric nozzle mechanism is the core part for thrust vectoring of aero engine, which contains complex rigid-flexible coupled multibody system with joints clearance and significantly reduces the efficiency in modeling and calculation, therefore the kinematics and dynamics analysis of axisymmetric vectoring nozzle mechanism based on deep neural network is proposed. The deep neural network model of the axisymmetric vector nozzle is established according to the limited training data from the physical dynamic model and then used to predict the kinematics and dynamics response of the axisymmetric vector nozzle. This study analyses the effects of joint clearance on the kinematics and dynamics of the axisymmetric vector nozzle mechanism by a data-driven model. It is found that the angular acceleration of the expanding blade and the driving force are mostly affected by joint clearance followed by the angle, angular velocity and position of the expanding blade. Larger joint clearance results in more pronounced fluctuations of the dynamic response of the mechanism, which is due to the greater relative velocity and contact force between the bushing and the pin. Since axisymmetric vector nozzles are highly complex nonlinear systems, traditional numerical methods of dynamics are extremely time-consuming. Our work indicates that the data-driven approach greatly reduces the computational cost while maintaining accuracy, and can be used for rapid evaluation and iterative computation of complex multibody dynamics of engine nozzle mechanism.

Advances and Prospects of PID Controller in Mechanical Field: From Traditional Tuning to Intelligent Optimization

Abstract. The extensive use of PID controllers in the mechanical fieldparticularly in robots and other mechanical systemsis covered in this study. In these sectors, PID controllers have become the most popular control technology due to their straightforward design and simplicity of use. PID controller performance in nonlinear complex systems is explored, along with stability and adaptive adjustment, by comparing the conventional and intelligent PID tuning methods. According to the research, control performance can be greatly enhanced by merging PID controller with intelligent technologies like fuzzy logic and genetic algorithms. These results point to the possible use of PID-based control in numerical control systems in the future. PID controllers' automatic tuning capabilities have garnered significant attention in the industrial sector. Future research will focus on the automatic tuning of PID controller to further optimize its performance and broaden its application range.

Adaptive MSSRCPF filtering based on constrained optimization and fading memory for SINS/GNSS integrated navigation

To address particle degeneracy and the challenge of selecting importance density functions in traditional particle filters for complex nonlinear systems, we propose an adaptive mixed-order spherical simplex-radial cubature particle filter algorithm based on constrained optimization and fading memory. This algorithm integrates constrained optimization, an adaptive fading memory strategy, adaptive adjustment of particle numbers and weights, and the advantages of mixed-order spherical simplex-radial cubature Kalman filtering. By designing the importance density function using a mixed-order integration method, the algorithm significantly improves filtering accuracy over traditional cubature Kalman filters while maintaining lower computational complexity than high-order cubature Kalman filter methods. The adaptive fading memory strategy dynamically adjusts the covariance matrix, enhancing sensitivity to current measurement information and reducing the influence of historical data. By dynamically adjusting the noise covariance matrix and constraining the ratio between the error covariance and measurement noise covariance, the algorithm improves the convergence speed and accuracy of state estimation. An adaptive particle number and weight adjustment strategy based on effective sample size and entropy regularization dynamically adjusts the number of particles to reduce computational complexity while ensuring filtering accuracy, and employs entropy regularization to suppress weight over-concentration, thereby reducing the impact of outliers on the filtering results. Simulation results demonstrate that in complex SINS/GNSS integrated navigation systems, especially under high-noise and nonlinear conditions, the proposed algorithm significantly improves positioning accuracy compared to the CPF method, with the maximum latitude error reduced by approximately 54.8%, the maximum longitude error reduced by approximately 40.5%, and the maximum altitude error reduced by approximately 68.3%. Compared to the FMCPF method, the proposed algorithm also shows clear advantages in positioning accuracy, convergence speed, and computational efficiency, validating its effectiveness and practicality in complex environments.

Numerical Assessment of MHD Tri-Hybrid Nanoparticles by Tangent Hyperbolic Model with Chemical Reaction: Thermodynamic Analysis

This paper investigates the influence of magneto-tangent hyperbolic nanofluid on the flow of a tri-hybrid nanoliquid consisting of [Formula: see text], and [Formula: see text] particles suspended in [Formula: see text]. The entropy production is encountered in this analysis. The fluid flows over a stretch sheet is considered. In addition, the energy equation also assumes the existence of a uniform heat source or sink and thermal radiation. Furthermore, the concentration equation emphasizes the chemical reaction. The current proposed model yields a set of nonlinear governing equations. The modeled formulation is transformed into a dimensionless system through the application of a suitable alteration. The complex nonlinear equation system was solved using the bvp4c through numerical methods. The main motive of this exploration is to emphasize the rate of heat and mass transfer in a flow of [Formula: see text], and [Formula: see text]-based hybrid nanofluid across a stretch sheet. The graphical study illustrates that Weissenberg number and magnetic field enhancement result in decreasing the velocity. But thermal layer, entropy production, and Bejan number are enhanced with larger values of Weissenberg number and magnetic field. This study focuses on different profiles with various flow parameters. Furthermore, we have compared the tri-hybrid nanofluid with the hybrid and mono nanofluid in all the figures and tabular format. Additionally, we have compared tri-hybrid, hybrid, and mono nanofluid using graphs for velocity, temperature, concentration, entropy production, and Bejan number.

Nonlinear analysis of compound pendulum model

Abstract This paper investigates the relationships among various nonlinear physi cal equations, with a particular focus on the standard form of the ϕ^4 equation. Based on the theoretical framework of the Jacobi elliptic function, an exact solution for the ϕ^4 equation is derived. A key innovation of this work is the discovery of the consistency between the ϕ^4 equation and the motion equation of the compound pendulum. By utilizing this correspondence, an exact solution for the compound pendulum equation is obtained, grounded in the Jacobi elliptic function theory. Compared to numerical methods, this solution provides higher accuracy and has the potential to be applied to
more complex nonlinear physical systems. This model can also be applied in areas such as vibration damping in building materials, mechanical system analysis, and spacecraft control.

An innovative dynamic observer for nonlinear interconnected systems with uncertainties

In this paper, an innovative dynamic observer is applied to complex nonlinear interconnected systems with matched and mismatched uncertainties. This dynamic observer can estimate system states, which cannot be achieved during the design process for nonlinear interconnected systems with uncertainties. The proposed method has great identification ability with small estimated errors for the states of nonlinear interconnected systems with matched and mismatched uncertainties. It should be pointed out that the considered uncertainties of nonlinear interconnected systems have general forms, which means that the proposed method can be effectively used in more generalised nonlinear interconnected systems. Finally, simulation results for the lateral flight control system are presented to ensure the effectiveness of this strategy.

Data-driven predictor of control-affine nonlinear dynamics: Finite discrete-time bilinear approximation of koopman operator

This paper introduces a novel and efficient data-driven approach for approximating a finite discrete-time bilinear model of control affine nonlinear dynamical systems with a tunable parameter that balances model dimension and prediction accuracy. An approximation of the Koopman operator based on the evolutions of the nonlinear system measurements used to lift a control-affine nonlinear system to a higher dimensional model. However, higher dimensional spaces can result in a long learning time and the curse of dimensionality in control analysis. The proposed approach addresses these challenges by introducing a convex optimization which identifies informative observable functions. This technique allows for the adjustment of a parameter to strike a balance between model dimension and accuracy in prediction. The main contribution of this study is to introduce a reduced dimensional bilinear model for a nonlinear complex system. This achievement is made possible by implementing convex sparse optimization, enabling the exploration of informative estimated Koopman eigenfunctions while minimizing the number of system measurements required. The optimization problem is solved using the alternating direction method of multipliers. The effectiveness of the proposed method is evaluated on three different nonlinear systems: a numerical nonlinear system, a Van der Pol oscillator, and a Duffing oscillator. In the last simulation, an estimation of the Koopman linear model is considered as a special case, and the policy iteration algorithm is employed to evaluate optimal control designed for different reduced-dimensional models.

An artificial neural network based approach for harmonic component prediction in a distribution line

With the increasing use of nonlinear devices in both generation and consumption of power, it is essential that we develop accurate and quick control for active filters to suppress harmonics. Time delays between input and output are catastrophic for such filters which rely on real-time operation. Artificial Neural Networks (ANNs) are capable of modeling complex nonlinear systems through adjustments in their learned parameters. Once properly trained, they can produce highly accurate predictions at an instantaneous time frame. Leveraging these qualities, various complex control systems may be replaced or aided by neural networks to provide quick and precise responses. This paper proposes an ANN-based approach for the prediction of individual harmonic components using minimal inputs. By extracting and analyzing the nature of harmonic component magnitudes obtained from the survey of a particular area through real-time measurements, a sequential pattern in their occurrence is observed. Various neural network architectures are trained using the collected data and their performances are evaluated. The best-performing model, whose losses are minimal, is then used to observe the harmonic cancellation for multiple unseen cases through a simplified simulation in hardware-in-the-loop. These neural network structures, which produce instantaneous and accurate outputs, are effective in harmonic filtering.

A comprehensive control paradigm for prescribed performance attainment in complex nonlinear systems

This paper presents a comprehensive control framework tailored for achieving Prescribed Performance Control (PPC) in the context of complex nonlinear systems, addressing multifaceted challenges prevalent in control design. The focus is on a control scenario characterized by the simultaneous presence of nonlinearity, external disturbances, time delay, and unknown control direction—issues that pose considerable obstacles for existing solutions. To surmount these challenges, our proposed approach integrates well-established techniques, including neural networks, Nussbaum-type gains, and adaptive control strategies within a unified control design framework. The specific technical challenge addressed in this work involves the effective management of these intricate complexities in Single-Input Single-Output (SISO) systems. Our contributions extend the theoretical foundations, presenting an ideal PPC control design and introducing two adaptive neural network-based control methods capable of accommodating both known and unknown control directions. Utilizing Lyapunov–Krasovskii functionals, we showcase a unique integration that surpasses a mere combination of individual techniques. This work advances the theoretical underpinnings of control engineering tailored for real-world scenarios. The proposed controller’s efficacy is validated through rigorous simulations and compared with recent results and benchmark PPC controllers, establishing its superiority in addressing the intricacies of complex control scenarios.

Position closed loop control of stepper motors based on arctangent function

The high-precision position closed-loop control of stepper motors usually use field-oriented control. This method requires high-precision models of motors, however stepper motors are complex nonlinear systems, and high-precision models are difficult to obtain. In order to solve this problem, a pulse frequency modulation controller based on arc tangent function is proposed in this paper. This algorithm does not need accurate mathematical model, and its turning parameters have clear physical significance, which can be calculated according to the control requirements. The experimental results show that when tracking the sinusoidal signal θref=450sin(0.628t), the tracking errors (peak-peak value) of the arctangent control algorithm is reduced by nearly 42% compared with the PI control algorithm, which provides a new idea for the position closed-loop control of stepper motors.

Fixed-/preassigned-time synchronization for delayed complex-valued neural networks with discontinuous activations.

Finite-time synchronization is a crucial phenomenon observed in nonlinear complex systems, the settling time in such a dynamic phenomenon is heavily depends on the initial states which may be unaccessible beforehand in the real world. Eliminating the dependence of the settling time on initial states leads to major advantage and convenience in practical applications. This paper is concerned with the fixed-/preassigned-time synchronization of delayed complex-valued neural networks(CVNNs) with discontinuous activations. By designing novel state feedback controllers, and with the help of Filippov regularization and inequality techniques, some new criteria for achieving fixed-/preassigned-time synchronization are established. The obtained theoretical results cover and supplement existing ones of the CVNNs with continuous activations. In addition, the upper-bound of the settling time is explicitly estimated. Finally, the validity of the theoretical results is supported by numerical simulations.

Iterative control framework with application to guidance and attitude control of rockets

Traditional control methods for nonlinear dynamical systems are predicated on verification of complex mathematical conditions related to the existence of a positive-definite Lyapunov function whose value must strictly decrease with time. Rigorous verification of Lyapunov conditions can be extremely difficult in real-world systems with high-dimensional and complex dynamics. In this paper, we present a novel control logic that can be readily applied to a general class of nonlinear systems irrespective of the complexities in their dynamics. The Iterative Control Framework (ICF) is designed to guarantee the convergence of the closed-loop system state to zero without a priori verification of Lyapunov-like conditions. The underlying computational routine runs in the background in real time and reconfigures the control vector at each time step in such a way that when the control input is applied to the system, the system trajectory reaches closer to the desired state. The technique is applicable to a broad class of complex nonlinear systems but is particularly suitable for systems inherently admitting control action of short duration such as missiles, rockets, satellites, and space vehicles. In this work, we focus on the application of ICF to guidance and attitude control of rockets and missiles where actuation is provided via single-use thrusters.

Intermittent dynamics identification and prediction from experimental data of discrete-mode semiconductor lasers by reservoir computing

Analysis of intermittent dynamics from experimental data is essential to promote the understanding of practical complex nonlinear systems and their underlying physical mechanisms. In this paper, reservoir computing enabled dynamics prediction, and identification of two types of intermittent switching using experimental data from discrete-mode semiconductor lasers are rigorously studied and demonstrated. The results show that, for the dynamics prediction task, both regular and irregular intermittent switching can be predicted reliably by reservoir computing, achieving the average normalized mean-square error of less than 0.015. Additionally, the impact of the number of virtual nodes in the reservoir layer, as well as the train-test split ratio on prediction performance, is explored. For the dynamic identification task, a 2-class classification test is adopted, and the corresponding binary accuracy is calculated to evaluate the identification performance. The results demonstrate that the accuracy of identifying both regular and irregular intermittent switching exceeds 0.996. Compared with the conventional amplitude threshold identification method, the reservoir computing-driven dynamics identification method exhibits superior accuracy, especially in the intermittent transient transition regions.

Balance Controller Design for Inverted Pendulum Considering Detail Reward Function and Two-Phase Learning Protocol

As a complex nonlinear system, the inverted pendulum (IP) system has the characteristics of asymmetry and instability. In this paper, the IP system is controlled by a learned deep neural network (DNN) that directly maps the system states to control commands in an end-to-end style. On the basis of deep reinforcement learning (DRL), the detail reward function (DRF) is designed to guide the DNN learning control strategy, which greatly enhances the pertinence and flexibility of the control. Moreover, a two-phase learning protocol (offline learning phase and online learning phase) is proposed to solve the “real gap” problem of the IP system. Firstly, the DNN learns the offline control strategy based on a simplified IP dynamic model and DRF. Then, a security controller is designed and used on the IP platform to optimize the DNN online. The experimental results demonstrate that the DNN has good robustness to model errors after secondary learning on the platform. When the length of the pendulum is reduced by 25% or increased by 25%, the steady-state error of the pendulum angle is less than 0.05 rad. The error is within the allowable range. The DNN is robust to changes in the length of the pendulum. The DRF and the two-phase learning protocol improve the adaptability of the controller to the complex and variable characteristics of the real platform and provide reference for other learning-based robot control problems.