Almost sure convergence of extremum seeking algorithm using stochastic perturbation

This paper presents an extremum seeking (ES) algorithm where the perturbation signal is a martingale difference sequence (m.d.s.) with a vanishing variance. The measurement noise at the plant output is modeled by a superposition of deterministic component, and a non-stationary colored noise signal. The optimizing set point of the uncertain reference-to-output equilibrium map is estimated by a stochastic approximation (SA)-type algorithm. The algorithm has a vanishing gain sequence dependent on the set point estimates. By utilizing powerful tools of the martingale convergence theory it is proved that with probability one the set point estimates converge to the optimizing equilibrium point, in spite of the presence of a measurement noise. This result is derived without requiring boundedness or any prior condition on the set point estimates.     

Conformational Motions and Functionally Key Residues for Vitamin B12 Transporter BtuCD–BtuF Revealed by Elastic Network Model with a Function-Related Internal Coordinate

BtuCD–BtuF from Escherichia coli is a binding protein-dependent adenosine triphosphate (ATP)-binding cassette (ABC) transporter system that uses the energy of ATP hydrolysis to transmit vitamin B12 across cellular membranes. Experimental studies have showed that during the transport cycle, the transporter undergoes conformational transitions between the “inward-facing” and “outward-facing” states, which results in the open–closed motions of the cytoplasmic gate of the transport channel. The opening–closing of the channel gate play critical roles for the function of the transporter, which enables the substrate vitamin B12 to be translocated into the cell. In the present work, the extent of opening of the cytoplasmic gate was chosen as a function-related internal coordinate. Then the mean-square fluctuation of the internal coordinate, as well as the cross-correlation between the displacement of the internal coordinate and the movement of each residue in the protein, were calculated based on the normal mode analysis of the elastic network model to analyze the function-related motions encoded in the structure of the system. In addition, the key residues important for the functional motions of the transporter were predicted by using a perturbation method. In order to facilitate the calculations, the internal coordinate was introduced as one of the axes of the coordinate space and the conventional Cartesian coordinate space was transformed into the internal/Cartesian space with linear approximation. All the calculations were carried out in this internal/Cartesian space. Our method can successfully identify the functional motions and key residues for the transporter BtuCD–BtuF, which are well consistent with the experimental observations.     

Simultaneous inversion for a diffusion coefficient and a spatially dependent source term in the SFADE

This paper deals with an inverse problem of determining a diffusion coefficient and a spatially dependent source term simultaneously in one-dimensional (1-D) space fractional advection–diffusion equation with final observations using the optimal perturbation regularization algorithm. An implicit finite difference scheme for solving the forward problem is set forth, and a fine estimation to the spectrum radius of the coefficient matrix of the difference scheme is given with which unconditional stability and convergence are proved. The simultaneous inversion problem is transformed to a minimization problem, and existence of solution to the minimum problem is proved by continuity of the input–output mapping. The optimal perturbation algorithm is introduced to solve the inverse problem, and numerical inversions are performed with the source function taking on different forms and the diffusion coefficient taking on different values, respectively. The inversion solutions give good approximations to the exact solutions demonstrating that the optimal perturbation algorithm with the Sigmoid-type regularization parameter is efficient for the simultaneous inversion problem in the space fractional diffusion equation.     

Convergence analysis and performance of an improved gravitational search algorithm

Gravitational search algorithm (GSA) has been shown to yield good performance for solving various optimization problems. However, it tends to suffer from premature convergence and loses the abilities of exploration and exploitation when solving complex problems. This paper presents an improved gravitational search algorithm (IGSA) that first employs chaotic perturbation operator and then considers memory strategy to overcome the aforementioned problems. The chaotic operator can enhance its global convergence to escape from local optima, and the memory strategy provides a faster convergence and shares individual's best fitness history to improve the exploitation ability. After that, convergence analysis of the proposed IGSA is presented based on discrete-time linear system theory and results show that IGSA is not only guaranteed to converge under the conditions, but can converge to the global optima with the probability 1. Finally, choice of reasonable parameters for IGSA is discussed on four typical benchmark test functions based on sensitivity analysis. Moreover, IGSA is tested against a suite of benchmark functions with excellent results and is compared to GA, PSO, HS, WDO, CFO, APO and other well-known GSA variants presented in the literatures. The results obtained show that IGSA converges faster than GSA and other heuristic algorithms investigated in this paper with higher global optimization performance.     

Adaptive sliding-mode voltage control for inverter operating in islanded mode in microgrid

Harmonic current caused by nonlinear loads and parametric variations of output filter of inverters make popular proportional–integral–derivative (PID) voltage controller far beyond excellent performance in case of microgrid operating in islanded mode. Motivated by this limitation, this paper proposes an adaptive sliding-mode controller (ASMC) to enhance disturbance-rejection performance of control system of islanded parallel inverters. And adaptive algorithms are designed to observe external disturbances and internal perturbation so as to guarantee the globe robustness of control system of inverter. The switching gain of control input is designed to be a time-varying value which effectively reduces undesirable chattering of control input signal. Simulating and experimental results are presented that the total harmonic distortion, chattering, and steady-state error of output voltage of islanded parallel inverters are effectively reduced and the dynamic performances and the capability of perturbation rejection of control system of inverter are effectively enhanced.     

Strong consistency of infinitesimal perturbation analysis for tandem queueing networks

Strong consistency of infinitesimal perturbation analysis for the sojourn times in a class of tandem queueing networks is proved. Service times at the queues are correlated, and they are affine functions of the variable parameters. Differentiability of the average sojourn times is not assumed, but proved. The analysis is not based on assumptions of regenerative cycles of the networks but on stability and ergodicity of the queueing processes involved. The proof of strong consistency is based on a set of abstract conditions, described in terms of properties of the sample performance functions. These conditions are first shown to be sufficient for strong consistency, and then their validity for the networks in question is proved.Research supported in part by the NSF under grants Nos. ECS85-15449 and CDR-8803012, under ONR contract nos. N00014-90-K-1093 and N00014-89-J-1023, and under Army contract no. DAAL-03-83-K-0171. This author is now with the Department of Manufacturing Engineering, Boston University, Boston, MA 02215.     

Topics in nonlinear wave theory

The remarkable exact solutions, which have been found for the Kortewegde Vries and similar equations, are discussed using a perturbation approach. This leads quickly to the known solutions for multisoliton interactions, and it is hoped that it will prove useful in extending results to other cases. Then a numerical method for computing solutions to nonlinear wave equations, developed by Fornberg, is mixed with various theoretical ideas to explore a variety of problems. The problems include interacting waves, the evolution of initial steps and wells, and wavetrain instabilities.     

采场气体非稳流问题的摄动方法

以多孔介质渗流理论为基础，视采场气体流动为非稳定流问题，将应用数学中的奇异摄动理论应用于采矿问题，为采矿工程问题的研究提供了新的手段；为矿井安全工程、特别是实现自动监控提供了理论了依据和计算方法，丰富了采场通风理论。结果表明：用非稳流模型来描述采场通风问题，可获得动态流动特征。对照以往方法，这种动态描述无疑可促进矿山安全问题的理论研究。     

形状修改结构静响应的快速重分析

提供了一种用于结构静响应快速分析的边界元摄动法。本方法把结构的形状改变看作其边界节点坐标在某一确定值邻域内的摄动，利用此边界元为基础的摄动理论，得到修改后结构的位移、面力、应力。给出了二阶摄动法与迭代摄动法的计算格式；并指出了其应用范围；算例证明了本方法的正确性和有效性。     

Horizon mapping: shadows for bump-mapped surfaces

Bump mapping produces realistic shading by perturbing normal vectors to a surface, but does not show the shadows that the bumps cast on nearby parts of the same surface. In this paper, these shadows are found from precomputed tables of horizon angles, listing, for each position entry, the elevation of the horizon in a sampled collection of directions. These tables are made for bumps on a standard flat surface, and then a transformation is developed so that the same tables can be used for an arbitrary curved parametrized surface patch. This necessitates a new method for scaling the bump size to the patch size. Incremental calculations can be used in a scan line algorithm for polygonal surface approximations. The errors in the bump shadows are discussed, as well as their anti-aliasing. (An earlier version of this article appeared as Max [10].)