An optimal control approach to the optical flow problem

The optical flow problem is reduced to an optimal control problem governed by a linear parabolic equation having the unknown velocity field (the optical flow) as drift term. This model is derived from a new assumption, that is, the brightness intensity is conserved on a moving pattern driven by a Gaussian stochastic process. The optimality conditions are deduced by a passage to the limit technique in an approximating optimal control problem introduced for a regularization purpose. Finally, the controller uniqueness is addressed.     

Gronwall inequality for hybrid systems

The Gronwall inequality, a well-known and useful result both for continuous-time and discrete-time signals, is extended to hybrid signals, namely those that combine continuous time and discrete time. An application of the result to establishing a bounded energy bounded state property for hybrid systems with inputs is provided.     

Tidal turbine array optimisation using the adjoint approach

Oceanic tides have the potential to yield a vast amount of renewable energy. Tidal stream generators are one of the key technologies for extracting and harnessing this potential. In order to extract an economically useful amount of power, hundreds of tidal turbines must typically be deployed in an array. This naturally leads to the question of how these turbines should be configured to extract the maximum possible power: the positioning and the individual tuning of the turbines could significantly influence the extracted power, and hence is of major economic interest. However, manual optimisation is difficult due to legal site constraints, nonlinear interactions of the turbine wakes, and the cubic dependence of the power on the flow speed. The novel contribution of this paper is the formulation of this problem as an optimisation problem constrained by a physical model, which is then solved using an efficient gradient-based optimisation algorithm. In each optimisation iteration, a two-dimensional finite element shallow water model predicts the flow and the performance of the current array configuration. The gradient of the power extracted with respect to the turbine positions and their tuning parameters is then computed in a fraction of the time taken for a flow solution by solving the associated adjoint equations. These equations propagate causality backwards through the computation, from the power extracted back to the turbine positions and the tuning parameters. This yields the gradient at a cost almost independent of the number of turbines, which is crucial for any practical application. The utility of the approach is demonstrated by optimising turbine arrays in four idealised scenarios and a more realistic case with up to 256 turbines in the Inner Sound of the Pentland Firth, Scotland.     

A simple solution method for the blade element momentum equations with guaranteed convergence

The blade element momentum (BEM) equations, though conceptually simple, can be challenging to solve reliably and efficiently with high precision. These requirements are particularly important for efficient rotor blade optimization that utilizes gradient‐based algorithms. Many solution approaches exist for numerically converging the axial and tangential induction factors. These methods all generally suffer from a lack of robustness in some regions of the rotor blade design space, or require significantly increased complexity to promote convergence. The approach described here allows for the BEM equations to be parameterized by one variable—the local inflow angle. This reduction is mathematically equivalent, but greatly simplifies the solution approach. Namely, it allows for the use of one‐dimensional root‐finding algorithms for which very robust and efficient algorithms exist. This paper also discusses an appropriate arrangement of the equation and corresponding bounds for the one‐dimensional search—intervals that bracket the solution and over which the function is well behaved. The result is a methodology for solving the BEM equations with guaranteed convergence and at a superlinear rate.Copyright © 2013 John Wiley & Sons, Ltd.     

Modeling and analysis of an over-constrained flexure-based compliant mechanism

Bridge-type micro-displacement amplifier with flexure hinges is a classic displacement amplification mechanism. Existing theoretic models cannot predict its amplification ratio and input stiffness accurately and make it very difficult to confirm the amplifier’s performance and error compensation by means of these models, which is very significant in ultra-precision positioning. This paper focuses on the development of design equations that can accurately calculate the ideal displacement amplification ratio and input stiffness of the amplifier based on the thought of statically indeterminate structure. Force Method, Maxwell–Mohr Method, principle of superposition and deformation compatibility are used together to establish uncanonical linear homogeneous equations. The analytical results are verified by FEA simulations. The influence of the geometric parameters on the amplifier performance is investigated. It is noted that amplifier performance is more sensitive to the longitudinal distance of flexure hinges. Besides, two same-sized amplifiers with the opposite output directions can be clearly differentiated by these equations.     

Particle filtering with analytically guided sampling

Particle filtering (PF) is a popular nonlinear estimation technique and has been widely used in a variety of applications such as target tracking. Within the PF framework, one critical design choice is the selection of the proposal distribution from which particles are drawn. In this paper, we advocate using as proposal distribution a Gaussian-mixture-based approximation of the posterior probability density function (pdf) after taking into account the most recent measurement. The novelty of our approach is that the parameters of each Gaussian used in the mixture are determined analytically to match the modes of the underlying unknown posterior pdf. As a result, particles are sampled along the most probable regions of the state space, hence reducing the probability of particle depletion. Based on the analytically determined proposal distribution, we introduce a novel PF, termed analytically guided sampling-based PF, which is validated in range-only and bearing-only target tracking.     

MAC finite difference scheme for Stokes equations with damping on non-uniform grids

In this paper, we consider the numerical methods for stationary Stokes equations with damping. The mark and cell(MAC) method has been applied to discretize the problem on non-uniform grids. We establish the LBB condition and the stability for the MAC scheme. Then we obtain the second order super-convergence in L2 norm for both velocity and pressure on non-uniform grids. We also obtain the second order convergence for some terms of H1 norm of the velocity, and the other terms of H1 norm are second order convergence on uniform grids. Numerical experiments using the MAC scheme show agreement of the numerical results with theoretical analysis.     

Comments on “Robust circularly orthogonal moment based on Chebyshev rational function” [Digit. Signal Process. 62 (2017) 249–258]

Comments on the recent work of Guo et al. [1] are presented, in order to correct the erroneous equations. The corrected equations of the Chebychev rational moments, for gray-level images, are presented. In addition, one numerical experiment is performed to ensure the validity of the corrected equations.     

The optimal control related to Riemannian manifolds and the viscosity solutions to Hamilton–Jacobi–Bellman equations

In this paper we study the optimal stochastic control problem for stochastic differential equations on Riemannian manifolds. The cost functional is specified by controlled backward stochastic differential equations in Euclidean space. Under some suitable assumptions, we conclude that the value function is the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation which is a fully nonlinear parabolic partial differential equation on Riemannian manifolds.     

区间集上的一元格蕴涵代数方程

把格蕴涵代数中滤子、素滤子、LI-理想、对偶原子和凸子格等概念拓展到区间集进行了重新定义,研究了三种基本的区间集上的一元格蕴涵代数方程,并给出了方程的可解性判断条件以及方程解集的若干性质.