Uniting local and global controllers for uncertain nonlinear systems: beyond global inverse optimality

This paper provides a solution to a new problem of global robust control for uncertain nonlinear systems. A new recursive design of stabilizing feedback control is proposed in which inverse optimality is achieved globally through the selection of generalized state-dependent scaling. The inverse optimal control law can always be designed such that its linearization is identical to linear optimal control, i.e. optimal control, for the linearized system with respect to a prescribed quadratic cost functional. Like other backstepping methods, this design is always successful for systems in strict-feedback form. The significance of the result stems from the fact that our controllers achieve desired level of ‘global’ robustness which is prescribed a priori. By uniting locally optimal robust control and global robust control with global inverse optimality, one can obtain global control laws with reasonable robustness without solving Hamilton–Jacobi equations directly.     

Optimization method for the inverse problem of reconstructing the source term in a parabolic equation

This work investigates the inverse problem of reconstructing a spacewise dependent heat source in the parabolic heat equation using a final temperature measurement. Such problem has important application in a large field of applied science. On the basis of the optimal control framework, the existence and necessary condition of the minimizer for the cost functional are established. The global uniqueness and stability of the minimizer are deduced from the necessary condition. The Landweber iteration algorithm is applied to the inverse problem and some numerical results are presented for various typical test examples.     

Optimal surface parameterization using inverse curvature map

Mesh parameterization is a fundamental technique in computer graphics. Our paper focuses on solving the problem of finding the best discrete conformal mapping that also minimizes area distortion. Firstly, we deduce an exact analytical differential formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. Our result shows the curvature map is invertible. Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Secondly, we formulate the task of computing conformal parameterizations with least area distortions as a constrained nonlinear optimization problem in curvature space. We deduce explicit conditions for the optima. Thirdly, we give an energy form to measure the area distortions, and show it has a unique global minimum. We use this to design an efficient algorithm, called free boundary curvature diffusion, which is guaranteed to converge to the global minimum. This result proves the common belief that optimal parameterization with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results and practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel parameterization applications are also introduced.     

Mathematical programs with complementarity constraints in the context of inverse optimal control for locomotion

In this paper an inverse optimal control problem in the form of a mathematical program with complementarity constraints (MPCC) is considered and numerical experiences are discussed. The inverse optimal control problem arises in the context of human navigation where the body is modelled as a dynamical system and it is assumed that the motions are optimally controlled with respect to an unknown cost function. The goal of the inversion is now to find a cost function within a given parametrized family of candidate cost functions such that the corresponding optimal motion minimizes the deviation from given data. MPCCs are known to be a challenging class of optimization problems typically violating all standard constraint qualifications (CQs). We show that under certain assumptions the resulting MPCC fulfills CQs for MPCCs being the basis for theory on MPCC optimality conditions and consequently for numerical solution techniques. Finally, numerical results are presented for the discretized inverse optimal control problem of locomotion using different solution techniques based on relaxation and lifting.     

The stable regulator problem and its inverse

This paper provides a detailed survey and extension of certain properties of the stable regulator problem: determinemininf{u} intmin{0}max{infty} x'Qx + u'u dtsubject todot{x} = Fx + Gu; x(0) = x_{0}; liminf{t rightarrow infty} x(t) = 0whereQis not necessarily sign definite. First, equivalence conditions recently given by Willems for the existence of the minimum are extended to include statements in terms of the Hamiltonian matrix and spectral factorization. This provides a precise relation between the time-domain and frequency-domain solutions to the problems. Second, the inverse problem of whether a given feedbacku = -Kxis optimal for someQis easily resolved, as is the redundancy problem of distinct Q1and Q2, resulting in the same optimal control.     

Discrete-time inverse optimal neural control for synchronous generators

This paper presents a robust inverse optimal neural control approach for stabilization of discrete-time uncertain nonlinear systems, which simultaneously minimizes a meaningful cost functional. A neural identifier scheme is used to model the uncertain system, and based on this neural model and an appropriate control Lyapunov function, then the robust inverse optimal neural controller is synthesized. Applicability of the proposed scheme is illustrated via simulation results for a synchronous generator model.     

最小费用充电站选址问题的分支定界算法

电动汽车的充电站选址问题是当前社会的热点问题,其实质是组合优化中经典的NP-hard问题。基于最小开设费用对充电站选址问题进行研究,首先对该问题进行了数学建模,进而研究了该问题的数学性质并给予相应的证明,利用这些性质减小问题的规模,从而降低问题的求解难度;然后设计了上下界子算法以及降阶子算法,基于这些子算法提出了一种可以快速缩小问题规模同时得到最优解的分支定界算法,降低了时间复杂度,同时可以对解空间进行大量剪枝加快求解速度;最后通过分析和求解一个示例来进一步阐述所提算法的原理和执行过程。     

A wave-splitting based optimization approach to multi-dimensional time-domain electromagnetic inverse problems

An optimization approach to a multi-dimensional electromagnetic inverse problem in the time domain is considered. Wave-splitting is integrated in the optimization algorithm. An analysis based on energy estimates is given to show that the wave-splitting provides the best linear combination of the fields for use in this type of inverse problem. The permittivity, permeability and conductivity are reconstructed by minimizing an objective functional. An explicit expression for the gradient of the objective functional is derived by introducing some auxiliary functions and using the Gauss theorem. The parameters are reconstructed by an iterative conjugate gradient algorithm. Numerical results are presented.     

Structured matrix algorithms for inverse scattering on the line

Abstract In this article the Marchenko integral equations leading to the solution of the inverse scattering problem for the 1-D Schr?dinger equation on the line are solved numerically. The linear system obtained by discretization has a structured matrix which allows one to apply FFT based techniques to solve the inverse scattering problem with minimal computational complexity. The numerical results agree with exact solutions when available. A proof of the convergence of the discretization scheme is given. Keywords Structured matrix systems, 1-D inverse scattering, Marchenko integral equation     

2D inverse problem in a distributed parameter system

In this paper, we present a numerical solution for an inverse heat problem to estimate 2D space-wise coefficient in a parabolic system describing heat transfer. The inverse problem is recast into an optimization problem solved with a conjugate gradient method. This approach use the non linear optimization formulation by minimizing a cost function taking into account both the measured outputs of the system and the outputs calculated by means of the model. The solution is performed by using the free finite element software FreeFem. Numerical example is carried out to check the validity of the proposed method.     

Inverse optimality in the class of Hopfield neural networks with input nonlinearity

This paper presents the chaos suppression problem in the class of Hopfield neural networks (HNNs) with input nonlinearity using inverse optimality approach. Using the inverse optimality technique and based on Lyapunov stability theory, a stabilizing control law, which is optimal with respect to meaningful cost functional, is determined to achieve global asymptotically stability in the closed-loop system. Numerical simulation is performed on a four-dimensional hyper-chaotic HNN to demonstrate the effectiveness of the proposed method.     

组合优化问题反问题的研究进展

本文重点介绍了组合优化问题反问题的研究进展。具体内容包括：线性规划问题反问题、最短路问题反问题、最小费用流问题反问题和网络容量扩充问题反问题的提出背景、研究成果、应用前景及一些可能的研究方向。     

An effective L-MONG algorithm for solving multi-objective flow-shop inverse scheduling problems

Generally, in handling traditional scheduling problems, ideal manufacturing system environments are assumed before determining effective scheduling. Unfortunately, “ideal environments” are not always possible. Real systems often encounter some uncertainties which will change the status of manufacturing systems. These may cause the original schedule to no longer to be optimal or even feasible. Traditional scheduling methods are not effective in coping with these cases. Therefore, a new scheduling strategy called “inverse scheduling” has been proposed to handle these problems. To the best of our knowledge, this research is the first to provide a comprehensive mathematical model for multi-objective permutation flow-shop inverse scheduling problem (PFISP). In this paper, first, a PFISP mathematical model is devised and an effective hybrid multi-objective evolutionary algorithm is proposed to handle uncertain processing parameters (uncertainties) and multiple objectives at the same time. In the proposed algorithm, we take an insert method NEH-based (Nawaz–Enscore–Ham) as a local improving procedure and propose several adaptations including efficient initialization, decimal system encoding, elitism and population diversity. Finally, 119 public problem instances with different scales and statistical performance comparisons are provided for the proposed algorithm. The results show that the proposed algorithm performs better than the traditional multi-objective evolution algorithm (MOEA) in terms of searching quality, diversity level and efficiency. This paper is the first to propose a mathematical model and develop a hybrid MOEA algorithm to solve PFISP in inverse scheduling domain.     

A method of solving the coefficient inverse problems of wave tomography

In this paper, the problem of nonlinear wave tomography is formulated as a coefficient inverse problem for a hyperbolic equation in the time domain. Efficient methods for solving the inverse problems of wave tomography for the case of transparent boundary conditions are presented. The algorithms are designed for supercomputers. We prove the Fréchet differentiability theorem for the residual functional and derive an exact expression for the Fréchet derivative in the case of a transparent boundary in the direct and conjugate problems. The expression for the Fréchet derivative of the residual functional remains valid if the experimental data are provided for only a part of the boundary. The effectiveness of the proposed method is illustrated by the numerical solution of a model problem of low-frequency wave tomography. The model problem is tailored to apply to the differential diagnosis of breast cancer.     

Iterative solvers for Tikhonov regularization of dense inverse problems

According to the special demands arising from the development of science and technology, in the last decades appeared a special class of problems that are inverse to the classical direct ones. Such an inverse problem is concerned with the opposite way, usually followed by a direct one: finding the cause of a given effect or finding the law of evolution given the cause and effect. Very frequently, such inverse problems are modelled by Fredholm first-kind integral equations that give rise after discretization to (very) ill-conditioned linear systems, in classical or least squares formulation. Then, an efficient numerical solution can be obtained by using the Tikhonov regularization technique. In this respect, in the present paper, we propose three Kovarik-like algorithms for numerical solution of the regularized problem. We prove convergence for all three methods and present numerical experiments on a mathematical model of an inverse problem concerned with the determination of charge distribution generating a given electric field.     

An evolutionary algorithm for multi-criteria inverse optimal value problems using a bilevel optimization model

Given a linear program, a desired optimal objective value, and a set of feasible cost vectors, one needs to determine a cost vector of the linear program such that the corresponding optimal objective value is closest to the desired value. The problem is always known as a standard inverse optimal value problem. When multiple criteria are adopted to determine cost vectors, a multi-criteria inverse optimal value problem arises, which is more general than the standard case. This paper focuses on the algorithmic approach for this class of problems, and develops an evolutionary algorithm based on a dynamic weighted aggregation method. First, the original problem is converted into a bilevel program with multiple upper level objectives, in which the lower level problem is a linear program for each fixed cost vector. In addition, the potential bases of the lower level program are encoded as chromosomes, and the weighted sum of the upper level objectives is taken as a new optimization function, by which some potential nondominated solutions can be generated. In the design of the evolutionary algorithm some specified characteristics of the problem are well utilized, such as the optimality conditions. Some preliminary computational experiments are reported, which demonstrates that the proposed algorithm is efficient and robust.     

A solution to the inverse kinematics of redundant manipulators

A solution to the inverse kinematics is a set of joint coordinates which correspond to a given set of task space coordinates (position and orientation of end effector). For the class of kinematically redundant robots, the solution is generically nonunique such that special methods are required for obtaining a solution. The method addressed in the paper, introduced earlier and termed “generalized inverse,” is based on a certain partitioning of the Jacobian functional corresponding to a nonlinear relationship of the inverse kinematics type. The article presents a new algorithm for solving the inverse kinematics using the method of generalized inverse based on a modified Newton-Raphson iterative technique. The new algorithm is efficient, converges rapidly, and completely generalizes the solution of the inverse kinematics problem for redundant robots. The method is illustrated by numerical examples.     

永磁同步电机调速系统的多维泰勒网逆控制

针对永磁同步电机(PMSM)的高性能控制问题,在充分考虑时变特性、不确定性以及测量噪声等随机因素的基础上,通过PMSM的逆系统将被控对象补偿成为具有线性传递关系的系统,提出一种基于改进自适应逆控制的控制方案.采用矢量控制的双闭环控制结构,将多维泰勒网逆控制方法引入速度环.首先,对PMSM数学模型的可逆性进行证明以解决非线性系统逆建模的存在性问题;然后,建立新颖的动态网络化控制器-----多维泰勒网(MTN),其具有结构简单、计算复杂度低的优点;最后,为了实现高精度的速度控制,将3个MTN分别作为实现系统建模的自适应模型辨识器、逆建模的自适应逆控制器和噪声干扰消除的非线性自适应滤波器,并将PMSM的动态响应控制和消除干扰的控制分为相对独立的过程进行,同时实现最优控制.仿真结果表明,所提出控制方案能够实现PMSM伺服系统精确的速度控制,具有良好的跟踪性能和较强的抗干扰能力.     

Stability in Probability and Inverse Optimal Control of Evolution Systems Driven by Lévy Processes

This paper first develops a Lyapunov-type theorem to study global well-posedness(existence and uniqueness of the strong variational solution)and asymptotic stability in probability of nonlinear stochastic evolution systems(SESs)driven by a special class of Levy processes,which consist of Wiener and compensated Poisson processes.This theorem is then utilized to develop an approach to solve an inverse optimal stabilization problem for SESs driven by Levy processes.The inverse optimal control design achieves global well-posedness and global asymptotic stability of the closed-loop system,and minimizes a meaningful cost functional that penalizes both states and control.The approach does not require to solve a Hamilton-Jacobi-Bellman equation(HJBE).An optimal stabilization of the evolution of the frequency of a certain genetic character from the population is included to illustrate the theoretical developments.