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XuetangX

最优控制

Jordan University of Science and Technology via XuetangX

Overview


《最优控制》是控制类研究生主干课程,在学习最优控制前,应具备线性系统理论相关的知识。在学习最优控制后,你将了解到:
1,针对给定的控制对象,设计合理的最优性能指标。
2,如何求解给定函数的极值以及取得该极值的条件。
3,使用动态规划算法解决最短路径问题或者其他多级优化问题。
4,针对给定的最优性能指标,使用以变分法和庞特里亚金极小值原理建立最优控制问题的必要条件。
5,研究航天器轨道规划、船舶导航、汽车主动悬架控制、最优投资策略等实际案例的求解方法。
6,使用MATLAB协助求解函数优化和偏微分方程,通过仿真验证所设计的最优控制器的性能。 "Optimal Control" is a core course for graduate students in control studies. Prior to learning optimal control, one should have a foundational understanding of linear system theory. Upon completing the course, you will gain insights into:
1. Designing appropriate optimal performance criteria for a given control object.
2. Ability to find the extremum of a given function and the conditions for achieving this extremum.
3. Using dynamic programming algorithms to solve shortest path problems or other multi-stage optimization issues.
4. Establishing the necessary conditions for optimal control problems using variational methods and Pontryagin's minimum principle, based on given optimal performance criteria.
5. Exploring solution methods for practical cases such as spacecraft trajectory planning, ship navigation, automotive active suspension control, and optimal investment strategies.
6. Utilizing MATLAB to assist in function optimization and solving partial differential equations, and to verify the performance of the designed optimal controller through simulation.

Syllabus

  • 1 Introduction
    • 1.1 Introduction to Optimal Control
    • 1.2 Mathematical Model and Physical Constraints
    • 1.3 The Optimal Control Problem
  • 2 The Performance Measure
    • 2.1 Performance Measures For Optimal Control Problems
    • 2.2 Selecting A Performance Measure
  • 3 Dynamic Programming
    • 3.1 Principle of Optimality
    • 3.2 Principal of Optimality and its applications
    • 3.3 Dynamic Programming Applied to Optimal Control Problems
    • 3.4 Interpolation
    • 3.5 A Recurrence Relation of Dynamic Programming
    • 3.6 Computational Procedure for Solving Optimal Control Problems
    • 3.7 Characteristic of Dynamic Programming Solution
    • 3.8 Discrete Linear Regulator Problems
    • 3.9 The Hamilton-Jacobi-Bellman Equation
    • 3.10 Continuous Linear Regulator Problems
    • 3.11 The Hamiltonian-Jacobi-Bellman Equation – Some Observations
  • 4 Calculus of Variations
    • 4.1.1 Fundamental Concepts
    • 4.1.2 Increment and Variation of a Functional
    • 4.2.1 Functionals of a Single Function
    • 4.2.2 Functionals of a Single Function with Specified Final Time
    • 4.2.3 Functionals of a Single Function with Specified End Points
    • 4.2.4 Functionals of a Single Function with Both the Final Time and End Points Free
    • 4.3.1 Functionals Involving Several Independent Functions – Fixed End Points
    • 4.3.2 Functionals Involving Several Independent Functions – Free End Points
    • 4.4 Piecewise-Smooth Extremals
    • 4.5.1 Constrained Minimization of Functions
    • 4.5.2 Constrained Minimization of Functions - Point and Differential Equation Constraints
    • 4.5.3 Constrained Minimization of Functions - Isoperimetric Constraints
  • 5 The Variational Approach to Optimal Control Problems
    • 5.1.1 Necessary Conditions for Optimal Control
    • 5.1.2 Boundary Conditions for Optimal Control Problems with Fixed Final Time
    • 5.1.3 Boundary Conditions for Optimal Control Problems with Free Final Time
    • 5.1.4 Boundary Conditions for Optimal Control Example with Fixed Final Time
    • 5.2.1 Linear Regulator Problems
    • 5.2.2 Linear Tracking Problems
    • 5.3.1 Pontryagin’s Minimum Principle
    • 5.3.2 Example of Pontryagin’s Minimum Principle
    • 5.3.3 State Variable Inequality Constraints
    • 5.4.1 Minimum-Time Problems
    • 5.4.2 The Form of the Optimal Control for a Class of Minimum-Time Problems
    • 5.4.3 Minimum-Time Control of Time-invariant Linear Systems
    • 5.5.1 Minimum Control-Effort Problems
    • 5.5.2 Example of Minimum Control-Effort Problems
    • 5.5.3 Example of Minimum-Fuel Problems
    • 5.5.4 A Weighted Combination of Elapsed Time and Consumed Fuel as the Performance Measure
    • 5.5.6 Minimum-Energy Problems
    • 5.6.1 Singular Intervals in Linear Time-Optimal Problems
    • 5.6.2 Singular Intervals in Linear Fuel-Optimal Problems
    • 5.6.3 Effects of Singular Intervals on Problem Solution
  • 6 Case Study
    • 6.1 Optimal Control for Ship Navigation
    • 6.2 Optimal Control for Spaceship Orbit Transfer
    • 6.3 LQR Control for Active Suspension Systems
    • 6.4 Optimal Investment Strategy Design
  • Final Exam

    Taught by

    Yong-Wei Zhang

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