System and method for controlling an operation of a system subject to an uncertainty
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Solution Overview
Problem
Existing motion planning systems for autonomous devices face challenges in handling uncertainty due to unmodeled phenomena and sensor limitations, leading to computationally intractable chance-constrained optimization problems that require approximate formulations or significant computational resources.
Innovation Solution
A data-driven approach using empirical quantile functions and robust optimization to reformulate chance constraints into deterministic constraints, allowing for optimal motion trajectory planning without prior knowledge of uncertainty distribution, utilizing confidence bounds and off-the-shelf optimization solvers.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If chance-constrained optimization method is used to handle uncertainty, then reliability of constraint satisfaction is improved, but computational complexity increases making the problem intractable
Solution Approach 1:
The patent transforms the probabilistic constraint satisfaction problem into a deterministic optimization problem by changing the parameter representation from probability distributions to confidence bounds derived from empirical quantile functions. This parameter transformation allows the use of standard optimization solvers while maintaining the desired reliability level through the confidence parameter δ.
Solution Approach 2:
The patent replaces the complex probabilistic constraint mechanism with a deterministic equivalent using robust optimization. By substituting the chance constraint formulation with a deterministic optimization framework that uses confidence bounds, the method eliminates the need for complex probabilistic computations while preserving the reliability guarantee.
2Device complexity
If data-driven approaches with scenarios are used to reformulate chance constraints, then computational tractability is improved, but the requirement for convex functions restricts the class of problems solvable
Solution Approach 1:
The patent changes the parameter representation from scenario-based probabilistic constraints to deterministic constraints using confidence bounds. This parameter transformation allows the method to handle non-convex functions by working directly with the empirical quantile function and its confidence bounds, thereby expanding the class of solvable problems beyond convex optimization.
3Adaptability or versatility
If mixed-integer formulation is used for data-driven approaches, then ability to handle non-convex problems is improved, but severe numerical challenges arise in implementation
Solution Approach 1:
The patent substitutes the mixed-integer formulation with a deterministic optimization approach using confidence bounds from empirical quantile functions. This substitution replaces the numerically challenging mixed-integer programming with a more stable deterministic framework that can handle non-convex problems through the use of robust optimization techniques and confidence interval theory.
4Adaptability or versatility
If multiple separate scenario problems are solved, then flexibility in handling different uncertainty realizations is improved, but significant computational resources are required
Solution Approach 1:
The patent merges multiple separate scenario problems into a single unified deterministic optimization problem. By combining the handling of all uncertainty realizations into one optimization framework that uses confidence bounds from the empirical quantile function, the method achieves the flexibility of handling multiple scenarios while improving computational efficiency through a single problem formulation rather than multiple separate solves.
Data Source
AI summary
The present disclosure discloses a system and a method for controlling an operation of a system subject to an uncertainty of an operation variable of the system. The method comprises collecting a number of samples of the uncertainty of the operation variable, constructing, based on the collected samples, an empirical quantile function associated with the uncertainty of the operation variable, determining confidence bounds on the empirical quantile function to bound an approximation error between the empirical quantile function and a true quantile function, determining an uncertainty set based on the empirical quantile function bounded by the confidence bounds, reformulating, based on the uncertainty set, a chance constraint into a deterministic constraint, solving an optimal control problem subject to the deterministic constraint to produce one or more control commands to one or more actuators of the system, and controlling the operation of the system based on the control commands.