Control of Chaotic Systems with Uncertainties by Orthogonal Function Neural Network

Backpropagation neural network has been applied successfully to solving uncertain problems in many fields. However, unsolved drawbacks still exist such as the problems of local minimum, slow convergence speed, and the determination of initial weights and the number of processing elements. In this paper, we introduce a single-layer orthogonal neural network (ONN) that is developed based on orthogonal functions. Since the processing elements are orthogonal to one another and there is no local minimum of the error function, the orthogonal neural network is able to avoid the above problems. Among the five existing orthogonal functions, Legendre polynomials and Chebyshev polynomials of the first kind have the properties of recursion and completeness. They are the most suitable to generate the neural network. Some typical examples are given to show their performance in function approximation. The results show that ONN has excellent convergence performance. Moreover, ONN is capable of approximating the mathematic model of backpropagation neural network. Therefore, it should be able to be applied to various applications that backpropagation neural network is suitable to solve. © 2001 John Wiley & Sons, Inc.

The design of a new hybrid controller for fractional-order uncertain chaotic systems with unknown time-varying delays

This article discusses the application of orthogonal neural networks to detect collisions between multiple robot manipulators that work in an overlapped space. By applying an expansion/shrinkage algorithm, the problem of collision detection between arms is transformed into that among cylinders (or rectangular solids) and line segments. This mapping simplifies the collision detection problem and thus neural networks can be applied to solve it. The property of parallel processing enables neural networks to detect collisions rapidly. A single‐layer orthogonal neural network is developed to avert the problems of conventional multilayer feedforward neural networks such as initial weights and the number of layers and processing elements. This orthogonal neural network can approximate various functions and is used to calculate forward solution and to detect collisions. An efficient neural network system for collision detection is also developed. © 1995 John Wiley & Sons, Inc.

Application of three-dimensional orthogonal neural network to craniomaxillary reconstruction

In this brief, an adaptive chaos control method is developed for stabilizing chaotic systems at their unknown equilibrium(s) using the invariant manifold theory. The developed method overcomes the problem that the equilibrium(s) of the chaotic systems are dependent on the unknown system parameters, which makes direct application of the conventional adaptive control difficult. Further development of the adaptive chaos control is undertaken for the situation where the parameter estimates are only allowed to vary within a bounded set due to the sensitivity of chaotic systems to parameter variations. A sufficient condition for convergence of system states and parameter estimates is obtained. The design method developed then is applied to stabilizing the Lorenz chaotic system at an unknown equilibrium. Both mathematical and computational results have demonstrated the effectiveness of this method.

In recent years, chaos synchronization has attracted many researchers’ interests. For a class of chaotic synchronization systems with unknown uncertainties caused by both model variations and external disturbances, an orthogonal function neural network is utilized to realize the synchronization of chaotic systems. The basis functions of orthogonal function neural network are Laguerre polynomials. First of all, the orthogonal function neural network is trained to learn the uncertain information. Then, the parameters of Laguerre orthogonal neural network are adjusted to accomplish the synchronization of two chaotic systems with the perturbation by Lyapunov steady theorem. At last, the result of numerical example is shown to illustrate the validity of the proposed method.

This article presents a new command-filtered composite adaptive neural control scheme for uncertain nonlinear systems. Compared with existing works, this approach focuses on achieving finite-time convergent composite adaptive control for the higher-order nonlinear system with unknown nonlinearities, parameter uncertainties, and external disturbances. First, radial basis function neural networks (NNs) are utilized to approximate the unknown functions of the considered uncertain nonlinear system. By constructing the prediction errors from the serial-parallel nonsmooth estimation models, the prediction errors and the tracking errors are fused to update the weights of the NNs. Afterward, the composite adaptive neural backstepping control scheme is proposed via nonsmooth command filter and adaptive disturbance estimation techniques. The proposed control scheme ensures that high-precision tracking performances and NN approximation performances can be achieved simultaneously. Meanwhile, it can avoid the singularity problem in the finite-time backstepping framework. Moreover, it is proved that all signals in the closed-loop control system can be convergent in finite time. Finally, simulation results are given to illustrate the effectiveness of the proposed control scheme.

This research introduces an innovative control method designed for the synchronization and management of chaotic systems through the application of advanced neural network techniques. Specifically, a neural network-based sliding mode control framework is employed to enhance system stability and precision in synchronization tasks. Chaotic systems, particularly when arranged in master–slave configurations, exhibit behaviors that are highly sensitive to initial conditions and parameter variations, making them ideal candidates for the proposed approach. The core of this methodology leverages neural networks to estimate unknown nonlinear functions and dynamically adjust the control coefficients, ensuring high accuracy and adaptability in real-time. One of the key contributions of this study lies in addressing the complex issues of parametric uncertainty, external disturbances, and unmodeled dynamics that typically challenge conventional sliding mode control methods. By incorporating neural networks, the controller is equipped to effectively mitigate these uncertainties, ensuring robust performance even in the face of significant system variability. The adaptive nature of the control system allows for continuous adjustment, resulting in improved synchronization accuracy and faster convergence times. The stability and robustness of the proposed control system are rigorously proven using Lyapunov-based methods. Simulations show that synchronization of nonlinear chaotic systems occurs within 10 s, even under varying conditions. This efficiency demonstrates its practical applications in secure communications, biological systems, and power grids, marking a significant advancement in chaotic system control with broad industrial potential.

This paper deals with an adaptive robust tracking control using a multilayer neural network (NN) for a class of nonlinear dynamic systems with unknown time varying state delays. Typical adaptive NN backstepping controllers for uncertain nonlinear systems with time-delay give rise to computation complexity caused by the the repeated derivatives of virtual controllers and nonlinear functions. Moreover, the combined techniques usually result in only uniformly ultimately bounded (UUB) stability caused by the inherent NN approximation error. This paper presents a control scheme that uses an integral sliding mode control as a feedback term and an adaptive neural controller as a feedforward term based on the desired compensation adaptive law (DCAL) technique. First, we develop a new DCAL formulation which avoids the explosion of complexity caused by the general NN backstepping scheme to compensate for nonlinear system uncertainties, bounded system disturbances, and unknown state time delays. Then, using a Lyapunov-Krasovskii (LK) functional, it is shown that the proposed controller renders the class of uncertain nonlinear time-delay systems asymptotically stable.

A novel single fuzzy approximation based adaptive control for a class of uncertain strict-feedback discrete-time nonlinear systems

A novel direct adaptive fuzzy neural networks(FNNs)controller for a class of unc ertain nonlinear chaotic system is presented.Takagi-Sugeno fuzzy logic systems a re applied to estimate the unknown function of the chaotic system,then the param eters of adaptive FNNs control law are on-line tuned so that the tracking prob le m is realized,i.e.to force the solutions of the chaotic plant to follow a certai n given trajectory under the condition that all signals involved are uniformly b ounded.Simulation results show the validity of the obtained result.

In this paper, a model of orthogonal multiwavelets neural network ensemble is proposed. The neural network ensemble consists of component orthogonal multiwavelets neural networks where each component neural network is trained by back propagation (BP) algorithm and with orthogonal multiwavelets functions in the hidden layer. Due to the orthogonality of orthogonal multiwavelets functions, all the hidden nodes are orthogonal and all the component neural networks are orthogonal, which can reduce the redundancy and improve the prediction accuracy for the network. The experimental results demonstrate that the proposed neural network ensemble has better generalization performance than BP neural network ensemble.

Robust adaptive control of chaotic system with uncertainty was investigated in the presence of modeling error. The scheme of adaptation was based on identification estimates via dynamical neural networks. By using proposed nonlinear adaptive controller, the chaotic signal of the unknown system dynamics tends to be driven into a well controlled steady state. Moreover, the mathematical proof of stability properties of the system was guaranteed. Finally, simulation results have demonstrated the effectiveness of the proposed method through application on the Chen's chaotic system.

Most optimal control method aim at finding a control law to minimize the corresponding cost function, and the key problem is to solve the Hamilton–Jacobi–Bellman equation (HJBE). However, the solution of HJBE is hard and even unavailable to obtain for nonlinear system and, thus, the optimal control method can not be developed. To address this problem, an adaptive neural inverse optimal control method is proposed for nonlinear systems with external disturbances and actuator failures in this article. The cost function in this article is defined based on the controller and, therefore, once the controller is obtained, the corresponding cost function is minimized. Accordingly, the directly solution of HJBE is avoided. In more detail, the auxiliary system is constructed to develop the auxiliary controller, based on which the actual optimal fault-tolerant controller can be further obtained. The uncertainties in system dynamics are compensated by neural networks, and the unknown actuator failures are handled by adaptive control gain. It is proved that nonlinear system is input-to-state stabilizable (ISS) and the proposed controller can minimize the cost function that puts penalty on tracking error and system input. Two simulation examples are provided to demonstrate the effectiveness of the proposed control method.

Sector-condition-based results for adaptive control and synchronization of chaotic systems under input saturation