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Besides inheriting the properties of classical Bézier curves of degree n, the corresponding λ-Bézier curves have a good performance on adjusting their shapes by changing shape control parameter. Specially, in the case where the shape control parameter equals zero, the λ-Bézier curves degenerate to the classical Bézier curves. In this paper, the shape modification of λ-Bézier curves by constrained optimization of position and tangent vector is investigated. The definition and properties of λ-Bézier curves are given in detail, and the shape modification is implemented by optimizing perturbations of control points. At the same time, the explicit formulas of modifying control points and shape parameter are obtained by Lagrange multiplier method. Using this algorithm, λ-Bézier curves are modified to satisfy the specified constraints of position and tangent vector, meanwhile the shape-preserving property is still retained. In order to illustrate its ability on adjusting the shape of λ-Bézier curves, some curve design applications are discussed, which show that the proposed method is effective and easy to implement. [ABSTRACT FROM AUTHOR]

A fractional order PID (FOPID) controller, which is suitable for control system designing for being insensitive to the variation in system parameter, is proposed for hydroturbine governing system in the paper. The simultaneous optimization for several parameters of controller, that is, Ki, Kd, Kp, λ, and μ, is done by a recently developed metaheuristic nature-inspired algorithm, namely, the firefly algorithm (FA), for the first time, where the selecting, moving, attractiveness behavior between fireflies and updating of brightness, and decision range are studied in detail to simulate the optimization process. Investigation clearly reveals the advantages of the FOPID controller over the integer controllers in terms of reduced oscillations and settling time. The present work also explores the superiority of FA based optimization technique in finding optimal parameters of the controller. Further, convergence characteristics of the FA are compared with optimum integer order PID (IOPID) controller to justify its efficiency. What is more, analysis confirms the robustness of FOPID controller under isolated load operation conditions. [ABSTRACT FROM AUTHOR]

The teaching-learning-based optimization (TLBO) algorithm is a population-based optimization algorithm which is based on the effect of the influence of a teacher on the output of learners in a class. A variant of teaching-learning-based optimization (TLBO) algorithm with differential learning (DLTLBO) is proposed in the paper. In this method, DLTLBO utilizes a learning strategy based on neighborhood search of teacher phase in the standard TLBO to generate a new mutation vector, while utilizing a differential learning to generate another new mutation vector. Then DLTLBO employs the crossover operation to generate new solutions so as to increase the diversity of the population. By the integration of the local search and the global search, DLTLBO achieves a tradeoff between exploration and exploitation. To demonstrate the effectiveness of our approaches, 24 benchmark functions are used for simulating and testing. Moreover, DLTLBO is used for parameter estimation of digital IIR filter and experimental results show that DLTLBO is superior or comparable to other given algorithms for the employed examples. [ABSTRACT FROM AUTHOR]