A direct adaptive state-feedback controller is proposed for highly nonlinear systems. We consider uncertain or ill-defined nonaffine nonlinear systems and employ a neural network (NN) with flexible structure, i.e., an online variation of the number of neurons. The NN approximates and adaptively cancels an unknown plant nonlinearity. A control law and adaptive laws for the weights in the hidden layer and output layer of the NN are established so that the whole closed-loop system is stable in the sense of Lyapunov. Moreover, the tracking error is guaranteed to be uniformly asymptotically stable (UAS) rather than uniformly ultimately bounded (UUB) with the aid of an additional robustifying control term. The proposed control algorithm is relatively simple and requires no restrictive conditions on the design constants for the stability. The efficiency of the proposed scheme is shown through the simulation of a simple nonaffine nonlinear system.

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We present a study on four new star clusters discovered in the halo of the intriguing dwarf irregular galaxy NGC 6822 from a wide field survey covering 3 • × 3 • area carried out with MegaCam at Canada-France-Hawaii Telescope (CFHT). The star clusters have extended structures with half-light radii R h ≈ 7.5-14.0 pc, larger than typical Galactic globular clusters and other known globular clusters in NGC 6822. The integrated colors and color magnitude diagrams (CMD) of resolved stars suggest that the new star clusters are 2 -10 Gyr old and relatively metal poor with Z=0.0001-0.004 based on the comparison with theoretical models. The projected distance of each star cluster from the galaxy center ranges from 10.7 ′ (≈ 1.5 kpc) to 77 ′ (≈ 11 kpc), far beyond the optical body of the galaxy. Interestingly, the new star clusters are aligned along the elongated old stellar halo of NGC 6822, which is almost perpendicular to the HI gas distribution where young stellar populations exist. We also find that the colors and half-light radii of the new clusters are correlated with the galactocentric distance: clusters farther from the galaxy center are larger and bluer than those closer to the galaxy center. We discuss the stellar structure and evolution of NGC 6822 implied by these new extended star clusters in the halo. We also discuss the current status of observational and theoretical understandings regarding the origin of extended star clusters in NGC 6822 and other galaxies.

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