Tracking within a time interval on the basis of data supplied by finite observers

We solve the tracking control problem, in which one should bring a trajectory of a system of linear ordinary differential equations into a neighborhood of a trajectory of another system within a given time interval. After getting into this neighborhood, one should keep the trajectory of the first subsystem in it for a time interval of given duration. For the control synthesis, we use incomplete and imprecise information on the online deviation of one trajectory from the other, which is obtained in real time from linear equations of observation. We consider distinct structures of observers, which substantially affect the solution of control problems for such systems. The equations of dynamics and admissible measurements contain uncertainty for which one knows only some hard pointwise constraints. To solve the main problem, we use an approach that can be reduced to the construction of auxiliary information sets and weakly invariant sets with a subsequent “aiming” of one set at a tube. We suggest an efficient method for an approximate solution on the basis of ellipsoidal calculus techniques. The results of the algorithm operation are illustrated by an example of the solution of a tracking control problem for two fourth-order subsystems.     

Convergence analysis of a global optimization algorithm using stochastic differential equations

We establish the convergence of a stochastic global optimization algorithm for general non-convex, smooth functions. The algorithm follows the trajectory of an appropriately defined stochastic differential equation (SDE). In order to achieve feasibility of the trajectory we introduce information from the Lagrange multipliers into the SDE. The analysis is performed in two steps. We first give a characterization of a probability measure (Π) that is defined on the set of global minima of the problem. We then study the transition density associated with the augmented diffusion process and show that its weak limit is given by Π.     

Properties of Hamiltonian systems in the Pontryagin maximum principle for economic growth problems

We consider an optimal control problem with a functional defined by an improper integral. We study the concavity properties of the maximized Hamiltonian and analyze the Hamiltonian systems in the Pontryagin maximum principle. On the basis of this analysis, we propose an algorithm for constructing an optimal trajectory by gluing the dynamics of the Hamiltonian systems. The algorithm is illustrated by calculating an optimal economic growth trajectory for macroeconomic data.     

Trajectory reconstruction for an autonomous aircraft based on five-beam scanning

The problem of reconstructing the trajectory of an aircraft is considered. The reconstruction is based on heights measured in flight and on preliminary information about the underlying elevation field. A trajectory-reconstructing algorithm is described, and numerical results produced by it are presented.     

Impulsive optimal control model for the trajectory of horizontal wells

This paper presents an impulsive optimal control model for solving the optimal designing problem of the trajectory of horizontal wells. We take fully into account the effect of unknown disturbances in drilling. The optimal control problem can be converted into a nonlinear parametric optimization by integrating the state equation. We discuss here that the locally optimal solution depends in a continuous way on the parameters (disturbances) and utilize this property to propose a revised Hooke–Jeeves algorithm. The uniform design technique is incorporated into the revised Hooke–Jeeves algorithm to handle the multimodal objective function. The numerical simulation is in accordance with theoretical results. The numerical results illustrate the validity of the model and efficiency of the algorithm.     

Some properties of the Hessian of the logarithmic barrier function

More than twenty years ago, Murray and Lootsma showed that Hessian matrices of the logarithmic barrier function become increasingly ill-conditioned at points on the barrier trajectory as the solution is approached. This paper explores some further characteristics of the barrier Hessian. We first show that, except in two special cases, the barrier Hessian is ill-conditioned in an entire region near the solution. At points in a more restricted region (including the barrier trajectory itself), this ill-conditioning displays a special structure connected with subspaces defined by the Jacobian of the active constraints. We then indicate how a Cholesky factorization with diagonal pivoting can be used to detect numerical rank-deficiency in the barrier Hessian, and to provide information about the underlying subspaces without making an explicit prediction of the active constraints. Using this subspace information, a close approximation to the Newton direction can be calculated by solving linear systems whose condition reflects that of the original problem.     

Generating a reference trajectory with defined kinematics for the IRb-6 manipulator

The fundamental problem in industrial robots control concerns algorithms generating reference trajectories.References [1–4] suggest generating algorithms of a reference trajectory, which are based on an arbitrary discretization of the manipulator's internal coordinates. Each point of discretization in the external space approximating a reference trajectory corresponds to known discretized internal coordinates of the manipulator.In [5–7], iteration methods of determining the internal coordinates corresponding to external coordinates of the reference trajectory point have been suggested. In this method of internal coordinates, determining the point of the reference trajectory is being approached in successive steps of an iterative computation. In [5], a modified iterative method of generation of a straight segment reference trajectory has been presented.Analytic formulae, which are the solution of an inverse problem of manipulator kinematics, enable design of trajectory generating algorithms which compute, in one step only, the internal coordinates of points lying exactly on the reference trajectory, with the accuracy resulting from the computer register length.In this paper, the author has presented an original PLAN2 computer algorithm generating reference trajectories of motion for a task. The kinematics of those trajectories is defined at selected points through which a task is to be passed, the distances between them being optional. The algorithm is based on formulae which are analytic solutions to an inverse problem for an IRb-6 manipulator kinematics.     

On the convergence of SUMT

This paper investigates some convergence properties of Fiacco and McCormick's SUMT algorithm. A convex programming problem is given for which the SUMT algorithm generates a unique unconstrained minimizing trajectory having an infinite number of accumulation points, each a global minimizer to the original convex programming problem.     

基于多元线性回归分析和Chan算法的三维精确定位研究

基于无线通信基站的室内三维定位问题主要是在已知TOA和基站三维信息的情况计算出终端的位置信息,实际上就是求解方程组中的未知数.首先建立通用模型和算法求解出终端的定位,接着通过优化得到使用尽量少的基站数目实现近似最优精度的定位,然后利用提出的算法实现终端移动轨迹的二维定位,最后结合实际情况,考虑了基站的有效测量距离并设计了有效基站识别算法对问题进行了分析.     

基于多项式实数根的三维井眼轨道设计理论

三维井眼轨道设计问题需要求解多元非线性方程组,由于未知数多、方程的非线性强,一般难以求出解析解,通常使用数值迭代方法求数值解.对三维s型轨道设计问题依据已知设计参数进行了分类,发现了一套有效的数学化简技巧,求出了第1类初值问题的解析解和第Ⅱ-Ⅳ类初值问题的拟解析解.提出了轨道设计问题的特征多项式的新概念,并证明了轨道设计问题是否有解取决于特征多项式是否有实数根,解的个数不多于实数根的个数或个数的二倍.所提出的基于特征多项式实数根的拟解析算法对于求解轨道设计问题具有计算速度快、计算可靠性高、易于计算机编程实现等优点,在三维水平井轨道设计、三维绕障井轨道设计、防碰设计等方面具有比数值迭代方法更好的计算性能.     

Information Geometry and Interior-Point Algorithms in Semidefinite Programs and Symmetric Cone Programs

We develop an information geometric approach to conic programming. Information geometry is a differential geometric framework specifically tailored to deal with convexity, naturally arising in information science including statistics, machine learning and signal processing etc. First we introduce an information geometric framework of conic programming. Then we focus on semidefinite and symmetric cone programs. Recently, we demonstrated that the number of iterations of Mizuno–Todd–Ye predictor–corrector primal–dual interior-point methods is (asymptotically) expressed with an integral over the central trajectory called “the curvature integral”. The number of iterations of the algorithm is approximated surprisingly well with the integral even for fairly large linear/semidefinite programs with thousands of variables. Here we prove that “the curvature integral” admits a rigorous differential geometric expression based on information geometry. We also obtain an interesting information geometric global theorem on the central trajectory for linear programs. Together with the numerical evidence in the aforementioned work, we claim that “the number of iterations of the interior-point algorithm is expressed as a differential geometric quantity.”     

The tracking of the trajectory of a dynamical system

The problem of tracking the trajectory of a dynamical system, described by a vector differential equation, is considered. An algorithm for solving this problem, based on the Krasovskii extremal shift method, well-known in position control theory, is proposed.     

On an algorithm for dynamic reconstruction of the input

We consider the problem of dynamic reconstruction of the input in a system described by a vector differential equation and nonlinear in the state variable. We indicate an algorithm that is stable under information noises and computational errors and is aimed at infinite system operation time. The algorithm is based on the dynamic regularization method.     

Reconstruction of controls in exponentially stable linear systems subjected to small perturbations

The problem of the dynamical reconstruction of the variable input of an exponentially stable linear system subjected to small non-linear perturbations is considered. In the case of inaccurate observations of its phase trajectory, an algorithm for solving this problem is given, based on the method of control with a model. The algorithm is stable to data interference and computation errors. General constructions are illustrated by an example in which the problem of reconstructing the input of an oscillatory section is discussed.     

Resource-saving tracking problem with infinite time horizon

We consider the problem on the infinite-duration tracking of a prescribed trajectory of an inaccurately observed control system subjected to an unobservable dynamic disturbance. We construct a solution algorithm that is resource-saving in the sense that the control resources used for solving the problem for small noise values in the state observation channel are little different from the corresponding resources in the “ideal case” where the current values of the dynamic disturbance are available to direct observation.     

Clustering of search trajectory and its application to parameter tuning

This paper is concerned with automated classification of Combinatorial Optimization Problem instances for instance-specific parameter tuning purpose. We propose the CluPaTra Framework, a generic approach to CLUster instances based on similar PAtterns according to search TRAjectories and apply it on parameter tuning. The key idea is to use the search trajectory as a generic feature for clustering problem instances. The advantage of using search trajectory is that it can be obtained from any local-search based algorithm with small additional computation time. We explore and compare two different search trajectory representations, two sequence alignment techniques (to calculate similarities) as well as two well-known clustering methods. We report experiment results on two classical problems: Travelling Salesman Problem and Quadratic Assignment Problem and industrial case study.     

Using Efficient Anchoring Points for Generating Search Directions in Interior Multiobjective Linear Programming

This paper modifies the affine-scaling primal algorithm to multiobjective linear programming (MOLP) problems. The modification is based on generating search directions in the form of projected gradients augmented by search directions pointing toward what we refer to as anchoring points. These anchoring points are located on the boundary of the feasible region and, together with the current, interior, iterate, define a cone in which we make the next step towards a solution of the MOLP problem. These anchoring points can be generated in more than one way. In this paper we present an approach that generates efficient anchoring points where the choice of termination solution available to the decision maker at each iteration consists of a set of efficient solutions. This set of efficient solutions is being updated during the iterative process so that only the most preferred solutions are retained for future considerations. Current MOLP algorithms are simplex-based and make their progress toward the optimal solution by following an exterior trajectory along the vertices of the constraints polytope. Since the proposed algorithm makes its progress through the interior of the constraints polytope, there is no need for vertex information and, therefore, the search for an acceptable solution may prove less sensitive to problem size. We refer to the resulting class of MOLP algorithms that are based on the affine-scaling primal algorithm as affine-scaling interior multiobjective linear programming (ASIMOLP) algorithms.     

On the convergence of CQ algorithm with variable steps for the split equality problem

We investigate CQ algorithm for the split equality problem in Hilbert spaces. In such an algorithm, the selection of the step requires prior information on the matrix norms, which is not always possible in practice. In this paper, we propose a new way to select the step so that the implementation of the algorithm does not need any prior information of the matrix norms. In Hilbert spaces, we establish the weak convergence of the proposed method to a solution of the problem under weaker conditions than usual. Preliminary numerical experiments show that the efficiency of the proposed algorithm when it applies the variable step-size.     

GMG — A guaranteed global optimization algorithm: Application to remote sensing

We investigate the role of additional information in reducing the computational complexity of the global optimization problem (GOP). Following this approach, we develop GMG — an algorithm for finding the Global Minimum with a Guarantee. The new algorithm breaks up an originally continuous GOP into a discrete (grid) search problem followed by a descent problem. The discrete search identifies the basin of attraction of the global minimum after which the actual location of the minimizer is found upon applying a descent algorithm. The algorithm is first applied to the golf-course problem, which serves as a litmus test for its performance in the presence of both complete and degraded additional information. GMG is further assessed on a set of standard benchmark functions. We then illustrate the performance of the validated algorithm on a simple realization of the monocular passive ranging (MPR) problem in remote sensing, which consists of identifying the range of an airborne target (missile, plane, etc.) from its observed radiance. This inverse problem is set as a GOP whereby the difference between the observed and model predicted radiances is minimized over the possible ranges and atmospheric conditions. We solve the GOP using GMG and report on the performance of the algorithm.     

Constrained optimization with stochastic feasibility regions applied to vehicle path planning

In real-time trajectory planning for unmanned vehicles, on-board sensors, radars and other instruments are used to collect information on possible obstacles to be avoided and pathways to be followed. Since, in practice, observations of the sensors have measurement errors, the stochasticity of the data has to be incorporated into the models. In this paper, we consider using a genetic algorithm for the constrained optimization problem of finding the trajectory with minimum length between two locations, avoiding the obstacles on the way. To incorporate the variability of the sensor readings, we propose a modified genetic algorithm, addressing the stochasticity of the feasible regions. In this way, the probability that a possible solution in the search space, say x, is feasible can be derived from the random observations of obstacles and pathways, creating a real-time data learning algorithm. By building a confidence region from the observed data such that its border intersects with the solution point x, the level of the confidence region defines the probability that x is feasible. We propose using a smooth penalty function based on the Gaussian distribution, facilitating the borders of the feasible regions to be reached by the algorithm.