Emotional Fuzzy Sliding-Mode Control for Unknown Nonlinear Systems

[[abstract]]The brain emotional learning model can be implemented with a simple hardware and processor; however, the learning model cannot model the qualitative aspects of human knowledge. To solve this problem, a fuzzy-based emotional learning model (FELM) with structure and parameter learning is proposed. The membership functions and fuzzy rules can be learned through the derived learning scheme. Further, an emotional fuzzy sliding-mode control (EFSMC) system, which does not need the plant model, is proposed for unknown nonlinear systems. The EFSMC system is applied to an inverted pendulum and a chaotic synchronization. The simulation results with the use of EFSMC system demonstrate the feasibility of FELM learning procedure. The main contributions of this paper are (1) the FELM varies its structure dynamically with a simple computation; (2) the parameter learning imitates the role of emotions in mammalians brain; (3) by combining the advantage of nonsingular terminal sliding-mode control, the EFSMC system provides very high precision and finite-time control performance; (4) the system analysis is given in the sense of the gradient descent method.[[notice]]補正完

On Weyl’s asymptotics and remainder term for the orthogonal and unitary groups

We examine the asymptotics of the spectral counting function of a compact Riemannian manifold by Avakumovic (Math Z 65:327–344, [1]) and Hörmander (Acta Math 121:193–218, [15]) and show that for the scale of orthogonal and unitary groups SO(N), SU(N), U(N) and Spin(N) it is not sharp. While for negative sectional curvature improvements are possible and known, cf. e.g., Duistermaat and Guillemin (Invent Math 29:39–79, [7]), here, we give sharp and contrasting examples in the positive Ricci curvature case [non-negative for U(N)]. Furthermore here the improvements are sharp and quantitative relating to the dimension and rank of the group. We discuss the implications of these results on the closely related problem of closed geodesics and the length spectrum