基于扩张原理的控制改进方法
发布时间：2022-04-22 23:02
半个多世纪以来,最优控制理论已经成为研究各类最优控制问题、控制目标和求解方法的一个广阔领域.控制目标的不断复杂化必然导致其数学模型和求精确解及渐近解的复杂性.由于研究对象的复杂性以及数学公式和计算变得更加复杂,寻找解析解变的更加困难甚至不可能做到.因此,诸如数值方法的渐近方法可用来寻找问题的渐近解.通常情况下,迭代法被用作渐近方法.然而只有当方程具有合适的初始近似时,才能得到理想的解.由于尚未建立普遍的计算方法和对初始近似的恰当选择的一般化结果,因此在这方面的研究对于提高迭代过程的计算效率是非常重要的.本文中,对最优控制问题的渐近分析应用扩张原理、局部化原理、Krotov最优解的充分条件、退化问题理论和Krotov极小化极大原理.特别地,建立了全新有效的迭代控制改进方法和最优控制问题渐近分析的一般方法.全文共分为六章.第一章主要介绍本文的研究背景、研究动机和主要结果.第二章提出了一种基于扩张原理和局部化原理的迭代最优化抽象方案.将该方法应用于连续最优控制问题,并给出了一般的最优性充分条件.此外,考虑到一些在实践中有用的矫正,得到了迭代控制的改进算法.本文的结果推广和补充了其他涉及最优性...
【文章页数】：119 页
【学位级别】：博士
【文章目录】：
中文摘要
Abstract
1 Introduction
1.1 Background
1.2 Motivation
1.3 Dissertation outline
2 Local control improvement for continuous systems
2.1 General approach to approximate optimization
2.2 Optimal control problem
2.3 Control improvement algorithm
2.4 Modifications
2.5 Problems
2.5.1 Problem 1
2.5.2 Problem 2
3 Local control improvement for discrete systems
3.1 Problem formulation and preliminary results
3.2 Control improvement
3.2.1 Derivation of the improvement algorithm
3.2.2 Iterative improvement algorithm
3.3 Transformations to discrete-continuous systems
3.4 Problems
3.4.1 Problem 1
3.4.2 Problem 2. Parallel reaction problem
3.4.3 Problem 3
4 Multistage approximate control optimization procedure
4.1 Abstract optimization problem and principles of solving it
4.2 General optimal control problem for a continuous system
4.2.1 Constructing estimates for the boundaries of the admissible reachability set
4.3 Transforming a controllable differential system in the general form to a system with linear controls
4.3.1 Algorithm for convex hull construction
4.4 General scheme for an approximate study of the original problem
4.5 Problems
4.5.1 Problem 1
4.5.2 Problem 2
4.5.3 Problem 3. Optimal control for a medical process
4.5.4 Problem 4. Excitation transfer along a spin chain
5 Global control improvement
5.1 Abstract iterative optimization scheme
5.2 Global control improvement for continuous systems
5.3 Global control improvement for discrete systems
5.4 Different implementations of global improvement methods
5.5 Problems
5.5.1 Problem 1. Controlling a linear oscillator
5.5.2 Problem 2. Nonlinear nonconvex problem with special modes
5.5.3 Problem 3. Classical nonconvex quadratic problem
5.5.4 Problem 4. Case of a degenerate second variation of the functional
5.5.5 Problem 5. Optimizing the control over treating a viral disease based on a simple model of the immune process
5.5.6 Problem 6
6 Conclusions and expectations
Appendix
A1. The extension principle
A2. Derivative and limit systems
A3. Theorem 4.1
References
Acknowledgements
Resume
本文编号：3646849
【文章页数】：119 页
【学位级别】：博士
【文章目录】：
中文摘要
Abstract
1 Introduction
1.1 Background
1.2 Motivation
1.3 Dissertation outline
2 Local control improvement for continuous systems
2.1 General approach to approximate optimization
2.2 Optimal control problem
2.3 Control improvement algorithm
2.4 Modifications
2.5 Problems
2.5.1 Problem 1
2.5.2 Problem 2
3 Local control improvement for discrete systems
3.1 Problem formulation and preliminary results
3.2 Control improvement
3.2.1 Derivation of the improvement algorithm
3.2.2 Iterative improvement algorithm
3.3 Transformations to discrete-continuous systems
3.4 Problems
3.4.1 Problem 1
3.4.2 Problem 2. Parallel reaction problem
3.4.3 Problem 3
4 Multistage approximate control optimization procedure
4.1 Abstract optimization problem and principles of solving it
4.2 General optimal control problem for a continuous system
4.2.1 Constructing estimates for the boundaries of the admissible reachability set
4.3 Transforming a controllable differential system in the general form to a system with linear controls
4.3.1 Algorithm for convex hull construction
4.4 General scheme for an approximate study of the original problem
4.5 Problems
4.5.1 Problem 1
4.5.2 Problem 2
4.5.3 Problem 3. Optimal control for a medical process
4.5.4 Problem 4. Excitation transfer along a spin chain
5 Global control improvement
5.1 Abstract iterative optimization scheme
5.2 Global control improvement for continuous systems
5.3 Global control improvement for discrete systems
5.4 Different implementations of global improvement methods
5.5 Problems
5.5.1 Problem 1. Controlling a linear oscillator
5.5.2 Problem 2. Nonlinear nonconvex problem with special modes
5.5.3 Problem 3. Classical nonconvex quadratic problem
5.5.4 Problem 4. Case of a degenerate second variation of the functional
5.5.5 Problem 5. Optimizing the control over treating a viral disease based on a simple model of the immune process
5.5.6 Problem 6
6 Conclusions and expectations
Appendix
A1. The extension principle
A2. Derivative and limit systems
A3. Theorem 4.1
References
Acknowledgements
Resume
本文编号：3646849
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