Unknown Chaotic Systems Research Articles

Stabilizing machine learning prediction of dynamics: Novel noise-inspired regularization tested with reservoir computing

Recent work has shown that machine learning (ML) models can skillfully forecast the dynamics of unknown chaotic systems. Short-term predictions of the state evolution and long-term predictions of the statistical patterns of the dynamics (“climate”) can be produced by employing a feedback loop, whereby the model is trained to predict forward only one time step, then the model output is used as input for multiple time steps. In the absence of mitigating techniques, however, this feedback can result in artificially rapid error growth (“instability”). One established mitigating technique is to add noise to the ML model training input. Based on this technique, we formulate a new penalty term in the loss function for ML models with memory of past inputs that deterministically approximates the effect of many small, independent noise realizations added to the model input during training. We refer to this penalty and the resulting regularization as Linearized Multi-Noise Training (LMNT). We systematically examine the effect of LMNT, input noise, and other established regularization techniques in a case study using reservoir computing, a machine learning method using recurrent neural networks, to predict the spatiotemporal chaotic Kuramoto–Sivashinsky equation. We find that reservoir computers trained with noise or with LMNT produce climate predictions that appear to be indefinitely stable and have a climate very similar to the true system, while the short-term forecasts are substantially more accurate than those trained with other regularization techniques. Finally, we show the deterministic aspect of our LMNT regularization facilitates fast reservoir computer regularization hyperparameter tuning.

Fixed-Time Output-Constrained Synchronization of Unknown Chaotic Financial Systems Using Neural Learning

This article addresses the challenging problem of fixed-time output-constrained synchronization for master–slave chaotic financial systems with unknown parameters and perturbations. A fixed-time neural adaptive control approach is originally proposed with the aid of the barrier Lyapunov function (BLF) and neural network (NN) identification. The BLF is introduced to preserve the synchronization errors always within the predefined output constraints. The NN is adopted to identify the compound unknown item in the synchronization error system. Unlike the conventional NN identification, the concept of indirect NN identification is employed, and only a single adaptive learning parameter is required to be adjusted online. According to the stability argument, the proposed controller can ensure that all error variables in the closed-loop system regulate to the minor residual sets around zero in fixed time. Finally, simulations and comparisons are conducted to verify the efficiency and benefits of the proposed control strategy. It can be concluded from the simulation results that the proposed fixed-time neural adaptive controller is capable of achieving better synchronization performance than the compared linear feedback controller.

Second-order hyperparameter tuning of model-based and adaptive observers for time-varying and unknown chaotic systems

In this paper, a second-order hyperparameter tuning method is proposed to improve the performance of online gradient-descent optimization. Second-order gradient information of a cost function obtained from extremum seeking optimization is embedded into the adaptation of states and parameters. Thus, a faster adaptation capability is provided without computing the inverse Hessian matrix. The convergence property of the adaptation dynamics via proposed hyperparameter is shown using Lyapunov approach. The proposed hyperparameters and conventional learning rates are compared in numerical applications of model-based estimation and adaptive estimation as follows: i) model-based synchronization of chaotic Lü-systems with time-varying parameters is performed by using an efficient nonlinear observer, ii) an adaptive fuzzy neural-network observer based state estimation is conducted for unknown Duffing oscillator. In both cases, online gradient-descent adaptations are boosted using the proposed hyperparameter and conventional learning rates and their capabilities are measured in terms of root-mean squared-error performance. As a result, the proposed hyperparameter tuning method presented more accurate performances where application results are illustrated in figures and in a table.

Robust attitude control of the three-dimensional unknown chaotic satellite system

This paper proposes a novel continuous-time robust direct adaptive controller for the attitude control of the three-dimensional unknown chaotic spacecraft system. It considers that the plant’s nonlinear terms, exogenous disturbances, and model uncertainties are unknown and bounded; the controller design is independent of the system’s nonlinear terms. These controller attributes flourish the robust performance of the closed-loop and establish smooth state vector convergence to zero. The proposed controller consists of three parts: (1) a linear controller establishes the stability of the closed-loop at the origin, (2) a nonlinear controller component that autonomously adjusts the feedback gain, and (3) a nonlinear adaptive controller compensates for the model uncertainties and external disturbances using the online estimates of bounds and model uncertainties. The output of this part remains within a given upper and lower bound. The feedback controller gain is large when the state variables are away from the origin and become small in the origin’s vicinity. This feature is novel and contributes to the synthesis of smooth control effort that establishes robust fast and oscillation-free convergence of the state variables to zero. The Lyapunov direct stability analysis assures the global asymptotic robust stability of the closed-loop. Computer simulations and comparative analysis are included to verify the theoretical findings.

Evolutionary modeling algorithm of ordinary differential equations based on genetic modeling

Differential equations are often used to describe complex systems and nonlinear systems related to time, but it is difficult to establish an ideal model for such systems based on some observed data. Especially when dealing with unknown chaotic system data, it is blind and difficult to combine existing nonlinear analysis results with relevant experience. In this paper, through the study of genetic modeling, the optimization process of model parameters using GA is embedded in the optimization process of model structure using GP, and the local search process of neighborhood solution generated by GP-based standard mutation operator is carried out for some individuals in each evolution generation, and the evolution modeling algorithm of ordinary differential equations is designed and implemented, and an application example is given.

A Nonlinear Adaptive Controller for the Synchronization of Unknown Identical Chaotic Systems

This paper proposes a novel nonlinear adaptive controller for the synchronization of two identical unknown chaotic systems. The proposed controller accomplishes fast convergence of the steady-state error to the origin and reduces the amplitude of the oscillations in the error signals during the transient. The rapid convergence increases the disturbance rejection attributes of the controller. The reduction in the oscillations’ magnitude decreases energy consumption and lowers the actuators’ degradation, reducing the probability of failure. This controller consists of linear adaptive and nonlinear control components, and they jointly synthesize control signals to penalize the synchronization error. The direct linear adaptive term keeps the closed-loop system stable; it ensures that the synchronization error converges to zero. The nonlinear terms heavily penalize the large state error vector in the transient, and its effect becomes minimal in the steady-state. The linear adaptive control is active in the steady-state, which synthesizes a smooth control signal in the vicinity of zero; this behavior causes oscillation free convergence of the error. Analysis based on the Lyapunov direct theorem proves the global asymptotic stabilization of the closed-loop at the origin. The paper also introduces parameters update laws that estimate the controller and unknown parameters of the master system and assure the convergence. This work also presents a comparative study of a numerical example of two identical Lorenz-Like unknown chaotic systems.

A Wavelet-Based Robust Adaptive T-S Fuzzy Controller Design for Synchronization of Faulty Chaotic Gyrostat Systems

In this study, the synchronization problem of the faulty chaotic gyrostat master–slave systems using a wavelet-based robust adaptive T-S fuzzy control is investigated. This control system is comprised of a wavelet T-S fuzzy network (TSFN) controller, a feedback controller and a robust controller. Within this scheme, the wavelet TSFN is used in two separate approaches. At the first approach, it is utilized to approximate the nonlinear functions of the gyrostat systems, and at the second approach, it is used to approximate the lumped fault function of the control system. The feedback controller is developed to initially control the underlying system, and a robust controller is an adaptive controller which is used to dispel the effects of the approximation errors on the tracking performance. The parameters of wavelet TSFN and the bound parameter of the robust controller are tuned on-line by the derived adaptive laws based on Lyapunov stability analysis. The wavelet block acts as a feature extractor, reduces the number of fuzzy rules and also acts as a normalization block. The Lyapunov stability of the closed-loop system, robustness against the parameters uncertainties, the external disturbances and the approximation errors, as well as the convergence of the tracking errors and boundedness of all signals in the closed-loop system, are guaranteed. Finally, the synchronization problem of the unknown chaotic gyrostat systems in the presence of parameter uncertainties, external disturbances and actuator faults is simulated to illustrate the effectiveness of the proposed control scheme.

Robust Adaptive Multi-Switching Synchronization of Multiple Different Orders Unknown Chaotic Systems

Multi-switching synchronization (MSS) of multiple different orders unknown chaotic (UC) systems confines hacking in the digital transmission process. Similarly, the suppression of undesirable chattering increases synchronization performance. This paper proposes a new robust synchronization control (RASC) technique and discusses the MSS of multiple different orders UC systems. This controller accomplishes (i) quick convergence, (ii) reduces the transient oscillations, and (iii) the rate of convergence decreases in the vicinity of the origin that causes the suppression of chattering. Analysis based on the Lyapunov direct method assures this convergence behavior with any positive values of the feedback gains. This work also provides parameters updated law that estimates the true values of unknown parameters. Numerical examples of five UC systems different orders are simulated. The computer based graphical results validate the efficiency and performance of the proposed RASC technique and the synchronization strategy when compare to peer works. In the simulation, the proposed synchronization strategy successfully recovers an encrypted received image on a communication channel. The article suggests some future research problems to extend the use of the proposed work.

Adaptive combination synchronisation of unknown chaotic Lorenz, Lü, Rössler and Chen systems

This paper studies the adaptive combined synchronisation (ACS) of different unknown chaotic systems. In the proposed scheme, the state vectors of a response system synchronise with the corresponding state vectors of the combined dual drive systems. The Lyapunov stability theory proves the asymptotic stability of the closed loop system at the origin. This work provides parameter update laws that estimate the actual values of unknown parameters. Important numerical simulation is included to show the effectiveness of the propose synchronisation scheme.

Adaptive combination synchronisation of unknown chaotic Lorenz, L

This paper studies the adaptive combined synchronisation (ACS) of different unknown chaotic systems. In the proposed scheme, the state vectors of a response system synchronise with the corresponding state vectors of the combined dual drive systems. The Lyapunov stability theory proves the asymptotic stability of the closed loop system at the origin. This work provides parameter update laws that estimate the actual values of unknown parameters. Important numerical simulation is included to show the effectiveness of the propose synchronisation scheme.

Synchronization of chaotic systems and their machine-learning models.

Recent advances have demonstrated the effectiveness of a machine-learning approach known as "reservoir computing" for model-free prediction of chaotic systems. We find that a well-trained reservoir computer can synchronize with its learned chaotic systems by linking them with a common signal. A necessary condition for achieving this synchronization is the negative values of the sub-Lyapunov exponents. Remarkably, we show that by sending just a scalar signal, one can achieve synchronism in trained reservoir computers and a cascading synchronization among chaotic systems and their fitted reservoir computers. Moreover, we demonstrate that this synchronization is maintained even in the presence of a parameter mismatch. Our findings possibly provide a path for accurate production of all expected signals in unknown chaotic systems using just one observational measure.

Prescribed performance synchronization for uncertain chaotic systems with input saturation based on neural networks

In this paper, a prescribed performance adaptive neural network synchronization is investigated for a class of unknown chaotic systems in the presence of input saturation and external unknown disturbances. A prescribed performance function is employed to transform the constraint problem of chaotic synchronization control error into the problem of guaranteeing the boundedness of the transformed error. By introducing the Gaussian error function, the input saturation is handled. A neural network-based synchronization control scheme is then developed. Under the developed synchronization control scheme, the synchronization of uncertain chaotic systems is achieved with different initial conditions. Numerical simulation results further demonstrate the effectiveness of the proposed synchronization control scheme for unknown chaotic systems subject to external unknown disturbances and input saturation.

Parameter estimation of unknown fractional-order memristor-based chaotic systems by a hybrid artificial bee colony algorithm combined with differential evolution

In this paper, parameter estimation of unknown fractional-order memristor-based chaotic systems is concerned. Firstly, the parameter estimation is transformed into a multi-dimensional optimization problem, where the fractional orders are treated as independent variables. Then, a hybrid artificial bee colony algorithm combined with differential evolution and other searching mechanisms is put forward to solve the optimization problem. Finally, to demonstrate the effectiveness of the proposed method, numerical simulations based on two typical fractional-order memristor-based chaotic systems are conducted. The simulation results shows that the proposed approach for parameter estimation of unknown chaotic systems is a successful and promising method with higher calculation accuracy, faster convergence speed, and stronger robustness.

Off-policy integral reinforcement learning optimal tracking control for continuous-time chaotic systems**Project supported by the National Natural Science Foundation of China (Grant Nos. 61304079 and 61374105), the Beijing Natural Science Foundation, China (Grant Nos. 4132078 and 4143065), the China Postdoctoral Science Foundation (Grant No. 2013M530527), the Fundamental Research Funds for the Central Universities, China (Grant No. FRF-TP-14-119A2), and the Open Research

This paper estimates an off-policy integral reinforcement learning (IRL) algorithm to obtain the optimal tracking control of unknown chaotic systems. Off-policy IRL can learn the solution of the HJB equation from the system data generated by an arbitrary control. Moreover, off-policy IRL can be regarded as a direct learning method, which avoids the identification of system dynamics. In this paper, the performance index function is first given based on the system tracking error and control error. For solving the Hamilton–Jacobi–Bellman (HJB) equation, an off-policy IRL algorithm is proposed. It is proven that the iterative control makes the tracking error system asymptotically stable, and the iterative performance index function is convergent. Simulation study demonstrates the effectiveness of the developed tracking control method.

Adaptive stabilizer for chaotic networked systems with network-induced delays and packet losses

In this paper, an adaptive control approach is designed for compensating the delays, packet losses and nonlinearities in the networked chaotic systems and maintaining the acceptable system stability. A state-feedback adaptive control scheme for unknown chaotic multi-input systems is proposed. Based on Lyapunov–Krasovskii theory, a normalized adaptive control law is synthesized to accommodate delays and system nonlinearities. Illustrative examples are studied. The simulation results show the effectiveness of the design method.

Reconstruction of one-dimensional chaotic maps from sequences of probability density functions

In many practical situations, it is impossible to measure the individual trajectories generated by an unknown chaotic system, but we can observe the evolution of probability density functions generated by such a system. The paper proposes for the first time a matrix-based approach to solve the generalized inverse Frobenius–Perron problem, that is, to reconstruct an unknown one-dimensional chaotic transformation, based on a temporal sequence of probability density functions generated by the transformation. Numerical examples are used to demonstrate the applicability of the proposed approach and evaluate its robustness with respect to constantly applied stochastic perturbations.

The adaptive control of unknown chaotic systems with external disturbance via a single input

This paper investigates the control of chaotic systems in the presence of unknown system parameters and external disturbance. We first investigate the control of a class of special chaotic systems and then discuss the control of general chaotic systems. Some robust criteria are proposed based on adaptive control scheme. By introducing proper auxiliary variables, the stability of the closed-loop system is proved using Lyapunov stability theory. As an example to illustrate the application of the proposed method, the control of the R\(\ddot{o}\)ssler chaotic system is also investigated via a single input. Some numerical simulations are given to demonstrate the robustness and efficiency of the proposed approach.

Synchronization of Underactuated Unknown Heavy Symmetric Chaotic Gyroscopes via Optimal Gaussian Radial Basis Adaptive Variable Structure Control

This brief proposes modified projective synchronization (MPS) methods for underactuated unknown heavy symmetric chaotic gyroscope systems via optimal Gaussian radial basis adaptive variable structure control. Chaotic gyroscope systems are considered as underactuated systems where a control input is designed to synchronize the two degree of freedoms interactions. Until now, no investigation of this subject with one control input has been presented. The importance of obtaining synchronization objectives is specified when the dynamics of gyroscope system are unknown. In this brief, using the neural variable structure control technique, a control law is established that guarantees the MPS of underactuated unknown chaotic gyros. In the neural variable structure control, Gaussian radial basis functions are utilized to estimate online the system dynamic functions. Adaptation laws of the online estimator are derived in the sense of the Lyapunov function. Moreover, online and offline optimizers are applied to optimize the energy of the control signal. The proposed solution is generalized to chaos control of the mentioned gyroscopes. Numerical simulations are presented to verify the proposed synchronization methods.

Projective Synchronization for a Class of Unknown Chaotic Systems Based on Adaptive Fuzzy Control

Projective Synchronization for a Class of Unknown Chaotic Systems Based on Adaptive Fuzzy Control

Adaptive Synchronization of Unknown Chaotic Systems Using Mamdani Fuzzy Approach

The synchronization problem for a general-class of unknown chaotic systems is addressed. Fuzzy systems in Mamdani type are employed to provide an approximate model of the master chaotic system using an adaptive approach. Within this, a number of fuzzy systems are utilized to approximate the unknown nonlinear functions of the master system. An approximate model similar to the master system is constructed which is the slave system. The error dynamics between the master and slave chaotic systems is used to build a suitable control input and fuzzy systems’ parameters adaptive laws to force the slave system to be synchronized with master system. The stability of the overall synchronization system is derived based on Lyapunov stability theory. Its shown that under appropriate assumptions, the proposed approach guarantees the boundness and asymptotic convergence of synchronization errors to a small neighborhood of origin. An extensive simulation study is performed on both Duffing and Rössler chaotic systems to show the effectiveness of the proposed scheme.