Mixed Shooting Method for Multi-Point Boundary Value Problems
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Solution Overview
Problem
Solving multi-boundary value problems (MBVPs) in optimal control systems is challenging due to sensitivity of boundary conditions and coupling effects, leading to convergence issues and high computational complexity, especially in motion control systems with nonlinearities and large time intervals.
Innovation Solution
The mixed shooting method partitions parameters and boundary conditions into separate sets, allowing iterative determination of each set based on fixed values of the other, reducing the number of parameters and improving convergence by incorporating the structure of the optimal trajectory.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional shooting methods are used to solve MBVPs, then the control trajectory can be determined, but the processing time is excessive and convergence is difficult due to sensitivity of boundary conditions and coupling effects
Solution Approach 1:
The patent segments the coupled boundary value problem into multiple independent sub-problems by introducing intermediate boundary conditions at switching points. This segmentation allows each sub-problem to be solved separately with reduced computational complexity, avoiding the need to solve the entire coupled system simultaneously, thereby significantly reducing processing time while maintaining solution accuracy.
Solution Approach 2:
The patent performs preliminary classification of optimal control trajectories into different types based on the structure of differential equations and boundary conditions. By pre-identifying the trajectory type and corresponding differential equation structure before solving, the method eliminates the need for iterative guessing and trial-and-error approaches, directly leading to faster convergence and reduced processing time.
2Reliability
If iterative updating of initial values is used to solve boundary value problems, then the boundary conditions can be satisfied, but the convergence is slow and computational complexity is high
Solution Approach 1:
The patent performs preliminary classification of optimal control trajectories into different types based on the structure of differential equations and boundary conditions. By pre-identifying the trajectory type and corresponding differential equation structure before solving, the method eliminates the need for iterative guessing and trial-and-error approaches, directly leading to faster convergence and reduced computation time.
Solution Approach 2:
The patent uses feedback from the classification results to guide the solution process. By determining the trajectory type first, the method provides feedback that specifies which differential equations to use and what boundary conditions to apply, creating a directed solution path rather than random iteration, thereby improving computational efficiency while ensuring boundary condition satisfaction.
3Adaptability or versatility
If the number of parameters in MBVP is large, then the problem can represent complex motion constraints, but the difficulty of solving increases significantly
Solution Approach 1:
The patent segments the large-scale MBVP into multiple smaller sub-problems with fewer parameters each. By dividing the time interval and introducing intermediate boundaries, the method reduces the number of parameters that need to be solved simultaneously in each sub-problem, making the solution process more manageable while still capturing complex motion constraints through the combination of segmented solutions.
Solution Approach 2:
The patent performs preliminary classification to identify the structure of differential equations and boundary conditions before solving. This preliminary action reduces problem-solving complexity by pre-determining which parameters are relevant and how they should be organized, eliminating the need to handle all parameters equally and reducing the overall computational burden.
Data Source
AI summary
An operation of a system is controlled according to a control trajectory represented by a solution of a set of differential equations satisfying boundary conditions, wherein the set of differential equations includes ordered piecewise differential equations. The set of differential equations is parameterized to produce a first set of parameters representing values of the solution at switching times and a second set of parameters representing values of the switching times. The first and the second sets of parameters are determined alternately until the boundary conditions are satisfied to produce the solution; and the control trajectory is generated based on the solution to control the operation of the system.