Hamilton-Jacobi-Bellman Optimal Control via Anti-Diffusive Discretization
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Solution Overview
Problem
Current optimal control methods for deterministic systems, such as space launchers, often result in locally optimal solutions and require a priori knowledge of the trajectory, making it difficult to achieve globally optimal trajectories without significant computational resources and stability issues.
Innovation Solution
The method involves transposing the optimal control problem into Hamilton Jacobi Bellman equations, using an anti-diffusive discretization method like 'Ultra Bee' on spatial and temporal grids, and implementing sparse data structures to efficiently calculate the minimum time function and reconstruct the optimal control function, allowing for near real-time analysis and control.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If direct methods are used to optimize system trajectory, then computational simplicity is improved, but solution quality deteriorates (only locally optimal solutions are obtained)
Solution Approach 1:
The patent replaces traditional mechanical optimization approaches (direct methods using gradient descent or sequential quadratic programming) with a Hamilton-Jacobi-Bellman framework that transforms the optimal control problem into a partial differential equation. This substitution enables globally optimal solutions by leveraging the dynamic programming principle, where the value function satisfies the HJB equation, avoiding local optima inherent in gradient-based methods.
Solution Approach 2:
The patent performs preliminary discretization of the state space and time domain before solving the optimization problem. By pre-defining grids for state variables and time steps, the method transforms the continuous optimal control problem into a discrete dynamic programming problem, allowing systematic computation of globally optimal trajectories without requiring iterative local searches.
2Measurement precision
If indirect methods are used to optimize system trajectory, then solution quality is improved (globally optimal solutions can be obtained), but device complexity deteriorates (prior knowledge of trajectory structure is required)
Solution Approach 1:
The patent creates a simplified computational model by discretizing the continuous state space into grids and approximating the value function using these discrete representations. This copying approach transforms the complex continuous HJB PDE into a discrete dynamic programming problem that can be solved systematically, eliminating the need for prior trajectory structure knowledge while maintaining global optimality.
Solution Approach 2:
The patent segments the continuous state space and time domain into discrete grids, dividing the complex optimization problem into smaller, manageable subproblems. By segmenting the problem space, the method enables systematic computation of optimal policies through dynamic programming, where each grid point represents a subproblem that can be solved independently and combined to form the global solution.
3Measurement precision
If fine spatial and temporal grids are used to solve Hamilton-Jacobi-Bellman equations, then solution precision is improved, but computing resources deteriorate (memory and computation time increase)
Solution Approach 1:
The patent applies partial action by computing the value function only on relevant grid points rather than exhaustively solving the entire state space. By identifying and computing only the necessary portions of the solution domain, the method achieves high precision trajectories while significantly reducing memory requirements and computation time compared to full-state-space discretization.
Data Source
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AI summary
The method involves transpositioning a problem (P) into Hamilton Jacobi Bellman equations linking an integer of state variables and respective deputies, where the integer is a dimension of state space of physical system. The equations are digitally resolved by discretizing through an anti-diffusive process i.e. ultrabee process, on spatial and temporal grids representing a space of state and time. The grids are utilized by hollow data structures, and a minimal time function is calculated. Optimal trajectory is reconstructed from values obtained until calculation stage.