Today, four-dimensional chaotic systems are attracting considerable attention because of their special characteristics. This paper presents a non-equilibrium four-dimensional chaotic system with hidden attractors and investigates its dynamical behavior using a bifurcation diagram, as well as three well-known entropy measures, such as approximate entropy, sample entropy, and Fuzzy entropy. In order to stabilize the proposed chaotic system, an adaptive radial-basis function neural network (RBF-NN)–based control method is proposed to represent the model of the uncertain nonlinear dynamics of the system. The Lyapunov direct method-based stability analysis of the proposed approach guarantees that all of the closed-loop signals are semi-globally uniformly ultimately bounded. Also, adaptive learning laws are proposed to tune the weight coefficients of the RBF-NN. The proposed adaptive control approach requires neither the prior information about the uncertain dynamics nor the parameters value of the considered system. Results of simulation validate the performance of the proposed control method.

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This paper presents a novel fault detection and estimation (FDE) scheme for a class of Lipschitz nonlinear systems subjected to modeling and measurement uncertainties. Many works in this field assume that the states of the system are measurable and the fault function acts as an additive term. The proposed FDE strategy deals with the uncertain nonlinear systems subjected to multiplicative faults, and it does not rely on availability of the full state. Multiplicative fault corresponds to the parameter or structure changes in the system or in the process model. Also, actuator gain fault is another important fault type which can be modeled as a multiplicative fault. The effects of this kind of fault are combined with the inputs and outputs of the system in a multiplicative form which, in turn, make the detection and estimation of the fault complex. The proposed scheme is based on an adaptive diagnostic observer that not only can estimate the states of the system and generate the residual signal simultaneously, but also is able to estimate the characteristic and magnitude of an unknown fault. In the fault detection step, the threshold is derived analytically to ensure the robustness of the proposed detection scheme against uncertainties. Also in the fault estimation step, a robust adaptive law based on the switching σ -modification is developed to estimate the detected fault accurately. Design steps of the proposed estimation scheme are introduced in the form of LMI problem. This formulation provides an effective way to calculate the design parameters. The proposed FDE scheme guarantees that all signals are uniformly ultimately bounded. Simulation results of a single-link, flexible joint, robotic arm show the effectiveness and robustness of the proposed FDE strategy.

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