Typical Nonlinear System Research Articles

Research on Nonlinear Systems Modeling Methods Based on Neural Networks

Nonlinear systems widely exist in all fields of industrial production and are difficult to model because of complex non-linearity. Neural network is widely used in process prediction, fault detection and fault diagnosis of modern industry because of the nonlinear fitting ability. Due to various structures, there exists diversity in the performance of neural networks. However, only the appropriate network can improve the efficiency and safety in modelling nonlinear industrial process, which requires full consideration of the structure of neural network. In this study, several typical structures of neural networks are compared and analysed, and the performance differences caused by these structures are presented in detail. Finally, performance differences of neural networks with inconsistent structures are verified on several experiments. The results showed that neural networks with inconsistent structures were good at dealing with different types of nonlinear systems. Our work will provide a theoretical basis in accurately modeling the industrial production process, which is beneficial to nonlinear system control.

Direct tuning method of gain‐scheduled controllers with the sparse polynomials function

AbstractIn industry, gain‐scheduled proportional‐integral‐derivative (PID) control is performed for nonlinear systems using a look‐up table (LUT) that is easy to understand. Compared with the fixed PID, there are many more parameters of the scheduler, and it takes a lot of time to tune them. Also, the ROM storage area increases. To address these problems, in this paper, we propose a gain‐scheduled control law using the sparse polynomial functions and a direct parameter tuning method without system identification. The polynomial functions are used instead of LUT to reduce the ROM area. For direct tuning, data‐driven control is formulated so that it can be applied to the gain‐scheduled control, and the optimal parameters are obtained by the LASSO regression, with which the small contributing parameters of the scheduler become zero, and a sparse controller is obtained. The effectiveness of this method was examined by simulation for two types of nonlinear systems. As a result, it was revealed that a sparse controller with a low calculation cost and a reduced ROM area can be directly obtained without knowing the characteristics of the controlled object for a large number of control parameters of the gain scheduler.

The Multiplicative Base Solution to the Differential Equations in Analog Circuits

This work focuses to solve any order of scalar differential equation involved in analog circuit representation. These types of mathematical representations have many applications in analysis and processing such as noise, filter, audio, RLC distributed interconnection (nodes) and transmission lines. Such systems are represented with scalar type differential equations and use numerical method to find a solution. One of the most successful methods is the fourth-order Runge–Kutta. This study introduced a multiplicative version of Runge–Kutta (MRK4) method. The performance analysis of the MRK4 is examined based on the error analysis method. The MRK4 method is applied to solve equations representing the linear and the nonlinear type systems. Results indicate the MRK4 to be superior with respect to the RK4 method.

Cost-based recurrence analysis of conductance time series for gas–liquid two-phase flow system

Gas–liquid two-phase flows are frequently encountered in chemical and nuclear industries. The study of gas–liquid flow structures is of great significance for understanding the mechanisms of the flow pattern transition. In this paper, a direct-image multi-electrode conductance sensor (DMCS) was used to detect the structure information of vertical gas–liquid flows. Recurrence plot (RP) and cost-based recurrence plot (CBRP) are validated using typical nonlinear systems, i.e. Lorenz system and Hénon map, and used to analyze the signals collected by the DMCS. The results indicate that the determinism (DET) derived from the CBRP is sensitive to flow pattern evolution, and can also demonstrate the internal differences in the same flow patterns.

Response of laboratory plasma influenced by a feedback loop to external force

AbstractThis study experimentally examines the sensitivity of laboratory plasma influenced by a feedback loop to applied external force, with the objective of gaining insight into chaos control in plasma. The ionization waves in laboratory plasma are adopted as the medium of a typical nonlinear system. In the case of a system without the influence of feedback (defined as system A), the orbit of the chaotic system changes to a periodic one as the intensity of the applied external force increases. In the case of a system with the influence of feedback (defined as system B), the chaotic orbit changes to a periodic one by the application of external force with lower intensity (approximately two-thirds) compared to system A. This reveals that the feedback loop enhances the response of the system to external forces.

Application of an adaptive PID controller enhanced by a differential evolution algorithm for precise control of dissolved oxygen in recirculating aquaculture systems

In aquaculture, the dissolved oxygen (DO) content of a water body is important for the growth of aquatic products and thus needs to be precisely controlled. For this purpose, through an open-loop experiment of DO aeration under different airflow rates, empirical transfer function models that can fully describe the dynamic response relationship between aeration flow rate and DO content were established. Based on these models, this study proposed a differential evolution (DE) algorithm-optimised radial basis function (RBF) neural network proportional integral derivative (PID) controller (DE-RBF-PID). The proposed controller has two optimisation parts. The first part is devoted to finding the optimal initial parameters of PID using an improved DE algorithm. The second part utilises the powerful learning ability of the RBF neural network to adjust the PID parameters online, which can not only eliminate overshoots, but also improve the controller adaptability. The simulation results for a typical DO nonlinear control system demonstrated the superiority of the proposed DE-RBF-PID controller over conventional PID and RBF-PID controllers. Therefore, this controller can be applied for precise tracking control of DO in complex circulating aquaculture systems. • The transfer function models of the control process for DO were established. • An adaptive PID controller enhanced by DE algorithm was proposed. • This proposed controller was used for DO control in circulating aquaculture. • This work can provide theoretical basis and accumulate experience for practical application.

Nonlinear Normal Modes in a Two-Stage Isolator Using a Modified Finite-Element Galerkin Method

A modified Galerkin method is proposed to approximate the nonlinear normal modes in a new type of a two-stage isolator. Besides the displacement of payload and the force transmissibility of this typical nonlinear dynamic system, the nonlinear normal modes defined as invariant manifolds can provide more information about the nonlinear coupling between the system components when periodic motions corresponding to the normal modes of the system occur. The presented approach applies a combination of finite-element discretization and Fourier series expansion for the approximate invariant manifolds. A Galerkin projection of the governing equations for the approximate invariant manifolds yields a set of nonlinear algebraic equations in expansion coefficients, which can be solved numerically with a general choice of zero as initial guess for the cases in this work. The resultant approximate solutions for the invariant manifolds can accurately describe the nonlinear interactions between system components in periodic motions of the specific nonlinear normal modes. In addition, one can solve the invariant manifolds for an annular domain of interest directly by this method, without considering other domain that includes the origin of phase space.

On disturbance rejection control for nonlinear three‐revolute prismatic spherical ball and plate systems

AbstractThe three‐revolute prismatic spherical ball and plate system (BP) is a typical complex nonlinear uncertain system involving axial coupling, friction, and other nonlinear factors. It is difficult to identify the model for each factor and isolate its effect from other factors. In addition to such challenge, the measurement noise, which is beyond the capability of many traditional closed‐loop control methods, exists in the system. For this reason, an active disturbance‐rejection control scheme combining linear tracking differentiator is presented to tackle the problems in the BP system. The simulation and real system test are conducted in response to point‐to‐point trajectories, circular and rectangular loci. The results demonstrate that fast response with small steady error can be achieved by the proposed control method.GTOC:The paper provide an active disturbance‐rejection control scheme combining linear tracking differentiator for the BP system control, The simulation and real system test demonstrate that fast response with small steady error can be achieved by the proposed control method.

Adaptive Fuzzy Path Tracking Control for Mobile Robots with Unknown Control Direction

In order to synthesize controllers for wheeled mobile robots (WMRs), some design techniques are usually based on the assumption that the center of mass is at the center of the robot itself. Nevertheless, the exact position of the center of mass is not easy to measure, thus WMRs is a typical uncertain nonlinear system with unknown control direction. Based on the fast terminal sliding mode control, an adaptive fuzzy path tracking control scheme is proposed for mobile robots with unknown control direction. In this scheme, the fuzzy system is used to approximate unknown functions, and a robust controller is constructed to compensate for the approximation error. The Nussbaum-type functions are integrated into the robust controller to estimate the unknown control direction. It is proved that all the signals in the closed-loop system are bounded, and the tracking error converges to a small neighborhood of the origin in a limited time. The effectiveness of the proposed scheme is illustrated by a simulation example.

Attitude control of self-balancing vehicle based on sliding mode variable structure control

The two wheeled self balancing vehicle belongs to a typical nonlinear system. Aiming at the problem of attitude control stability of two wheeled self-balancing vehicle. A new sliding mode reaching law of integral sliding mode variable structure control was proposed. The feasibility and effetivenes of the controller are proved by theory and experiment. The experiment results show that the designed controller has remarkable advantages of steady state characteristic,fast dynamic response and strong robustness to external disturbance to compared with traditional PID controller.

Analytic target cascading with fuzzy uncertainties based on global sensitivity analysis for overall design of launch vehicle powered by hybrid rocket motor

An analytic target cascading (ATC) method with fuzzy uncertainties based on global sensitivity analysis is proposed to solve the optimization problem of typical multilevel multidisciplinary nonlinear system. The overall design of launch vehicle (LV) powered by hybrid rocket motors (HRMs) involves multiply disciplines, and is decomposed into two levels using an ATC framework. The fuzzy theory is applied to describe the uncertain design factors caused by decisions and cognition insufficiency. The rank correlation coefficient method (RCCM) based on feasible optimization solutions and the quadratic response surface method (QRSM) based on Latin hypercube sampling (LHS) are used for global sensitivity analyses of input uncertainty and model uncertainty, respectively. The multi-island genetic algorithm (MIGA) is adopted in all examples, and the known two-phase optimization method is used to verify the LV design results. The results show that global sensitivity analysis can significantly filter the fuzzy uncertain factors which have little influence on the responses. The ATC decomposition is applicable in solving the calculation burden caused by uncertainties. The fuzzy-based design optimization with ATC is more efficient than that with MDF, and gives more reliable and robust results than the deterministic ATC method.

A novel decoupling dynamic method with third-order accuracy and controllable dissipation

In this paper, an efficient explicit integration method with third-order accuracy and controllable dissipation properties is proposed, and a new accuracy criterion is presented to evaluate relative errors of integration methods. By using this method, a desirable accuracy for responses in the low-frequency domain can be attained, spurious vibrations in the high-frequency domain can be strongly suppressed. Moreover, for the proposed method, update values of the displacement and velocity do not depend on the new state. Hence, it is convenient for decoupling a multi-spring coupling dynamic system to achieve high computational efficiency. The standard formulations of the proposed method, including the second-order and third-order accuracy schemes, are derived by discussing the local truncation error, difference accuracy, and stability firstly. Subsequently, the accuracy properties of the proposed method, e.g., the period elongation and amplitude decay, are analyzed in comparison with other available state-of-the-art explicit integration methods. Additionally, the properties of the proposed method, including the accuracy, convergence, dissipation, efficiency, and nonlinearity, are evaluated by using three examples. More specifically, the accuracy and convergence are evaluated through an example with theoretical solutions; the dissipation and efficiency are investigated by analyzing a geodesic dome truss; and the effectiveness in terms of the nonlinearity is investigated via a typical nonlinear dynamic system.

Optimal H-infinity Integral Dynamic State Feedback Model Reference Controller Design for Nonlinear Systems

In this work, a new optimal and robust controller is proposed for nonlinear systems by combining the H-infinity control approach with the model reference control technique. It is carried out with the aid of integral dynamic state feedback control to achieve an adequate transient response by incorporating a reference model. The black hole optimization is used as a novel and efficient optimization algorithm to optimize the design approach. In addition, the proposed controller gain matrix is calculated by solving the H-infinity algebraic Riccati equation. Moreover, the stability properties of the controlled system are demonstrated using Lyapunov stability analysis. Two types of nonlinear systems are used as case studies to show the effectiveness of the proposed controller. Finally, the simulation results confirm that the proposed controller is very effective in enhancing the stability and performance of the nonlinear systems by compensating them and forcing their states to track the reference model states asymptotically with an acceptable and workable control action.

Stability Analysis upon High-Pressure Common Rail Fuel Injection System under Multiple Injection Modes

Fuel injection stability of high-pressure common rail fuel injection system (HPCRFIS) under multiple injection modes is a recognized technical problem. To explain it essentially, a numerical model based on bond graph method, which provides a standardized series of symbol system to describe various components in different physical fields as well as interactive effect analysis tool for different energy categories, was used to investigate the problem. As HPCRFIS is a typical nonlinear system with multi-physical coupling, it is very difficult to analyze system stability directly. Therefore, the system was linearized at typical moments. The system matrix for each typical moment was obtained thereby. To investigate the influential mechanism of system stability deeply, the variations of rank and eigenvalues distributions of state matrices were analyzed. The results show that the rank of state matrices changes abruptly when the needle or control valve moves. In terms of system stability, the oscillation characteristics are subject to the movement of needle or control valve. The ending of the former injection causes the system to show damping oscillation characteristics before the latter injection starts. The motion of control valve makes more contribution to system stability than that of needle, which could be concluded by Lyapunov stability criterion.

RETRACTED: Design of rotational inverted pendulum control based on second-order sliding mode

Rotational Inverted Pendulum is a typical nonlinear, unstable and strong coupling system. How to design a control system to restore the equilibrium state quickly and stably is a problem worth studying. High order super twisting sliding mode system has the advantages of traditional sliding mode control system. Super twisting system can elimination control chatter due to the discontinuities of control variable which contain in traditional sliding mode system. The super twisting system is designed. The saturation function replaces the sign function in the super twisting control function to improve the smoothness of the control variable. The simulation shows that this design has excellent precision. Inverted pendulum has smaller movement stroke than traditional super twisting system. Control chatter is also smaller than traditional super twisting system.

Adaptive neural network finite-time tracking quantized control for uncertain nonlinear systems with full-state constraints and applications to QUAVs

A novel adaptive neural network finite-time tracking control strategy is developed for a class of uncertain nonlinear systems with asymmetric time-varying full-state constraints and input quantization. With the help of an introduced TVABLF, all state variables are confined to predefined regions. Based on the backstepping method and NNs approximation technique, a smooth tracking controller and its adaptive laws are co-designed for the uncertain systems. The system effects caused by the input quantization is compensated by the proposed algorithm with nonlinear decomposition. By adopting the SGPFS theory, the tracking performances are ensured to be achieved in finite time. It is rigorously proved that the output of the system follows the specified trajectory in finite time, whilst the system state variables are constrained within asymmetric boundaries. QUAVs are typical nonlinear systems of 6-Dof, and the control input signal requires quantization. We apply the developed method to the controller design of uncertain QUAVs, and the simulation results verify the effectiveness of the main results.

Disturbance rejection control for PMSM using integral sliding mode based composite nonlinear feedback control with load observer

The low-speed high-torque permanent magnet synchronous motor (PMSM) drive system is a kind of typical nonlinear, strong-coupling and easy-parameter perturbation electromechanical coupling system. The control system is uncertainties and subject to unknown external interferences as well. In this paper, a disturbance rejection control method combining robust speed controller and load observer is proposed for low-speed high-torque PMSM. The robust speed controller combines the composite nonlinear feedback (CNF) which has advantage in improving the transient responsive performance and the integral sliding mode (ISM) advancing in improving system robustness. Subsequently, the effects of unknown external interferences are avoided by using a sliding mode observer (SMO), in which the chattering is reduced by introducing fuzzy control, and the observation is used for feed-forward compensation. The proposed robust speed controller solves the contradiction between the rapidity and overshoot of the traditional control method, and combines the load observer to compensate the influence of the load mutations and wide range of the load changes on the control system. Finally, the numerical simulation and experiments demonstrate that the proposed speed control method is able to achieve good transient performance in inhibiting system overshoot and reducing stable state error. Additionally, it successfully suppresses the influence of load disturbances and mutations, and shows the proposed method has better robustness.

Nonlinear modal interaction analysis and vibration characteristics of a francis hydro-turbine generator unit

The Francis hydro-turbine generator unit (FHTGU) is a typical nonlinear system with the coupling hydraulic, mechanical and electric subsystems. It is a challenge to understand the reasons for its operational failures because the major reason for failures involves complex interactions of the three subsystems. Subsystems’ model interaction with the method of normal forms has been well developed and investigated, overcoming the linear methods used in the FHTGU’s stability analysis. However, these methods have not to quantify higher-order terms in a mathematically accurate type to capture dynamic modal interactions between subsystems. Due to the accelerating expansion of hydropower stations, stability of FHTGU shows singular nonlinear oscillations and new methods have to be upgraded to cope with this new situation. In this study, the nonlinear modal method is introduced to analyze the dynamic modal interactions between subsystems, and results given by the different methods are compared to verify the method’s feasibility. The effect of the second order modes is quantified to investigate its effect on the dynamic characteristics of FHTGU, and the vibration characteristics affected by the wind generation system are also investigated. The result shows that the intensity of modes can be effectively reduced to satisfy the stable requirements. All of these results provide a theoretical guidance for the stable operation of FHTGUs.

A New Control Strategy for Bi-directional DC/DC Converter in DC Microgrid

The bi-directional DC / DC converter in DC microgrid is a typical nonlinear system which has large voltage disturbance during lead accumulator charging and discharging. In order to solve the problem of voltage disturbance, the linearization of the converter is realized by exact feedback linearization, and the sliding mode controller is designed by using exponential approximation law. The simulation results show that the method has fast response speed, strong anti-interference ability and good steady-state characteristics.

Brain Affective System Inspired Control Architecture: An Application to Nonlinear System

This paper presents a bio-inspired intelligent controller to enhance the performance of a nonlinear system. The controller is designed by capturing the emotional intelligence of mammalian brain mediated by the limbic system in which certain parts are responsible for generating emotions and these can be combined together as brain affective system inspired control architecture (BASIC). This controller has been used to cope with nonlinearities present in control applications. In this paper brain inspired control architecture is proposed while incorporating the sensory cortex explicitly in the architecture and the computational equations of various modules are suitably developed. The performance of the proposed controller is analyzed on a typical nonlinear system, namely, Permanent magnet synchronous motor for speed control, harmonics in stator phase currents and ripples produced in electromagnetic torque at different operating conditions. Its performance is compared with the existing control techniques in offline simulations as well as in real-time Hardware-in-loop environment. The results establish the computational efficiency, accuracy and robustness of the proposed controller.