Conceitos Básicos
Developing a framework for approximating the distribution of non-linear stochastic dynamical systems with formal guarantees of correctness.
Resumo
The article focuses on uncertainty propagation in non-linear stochastic dynamical systems using mixture models. It introduces a novel approach to approximate system distribution over time with tractable approximations and correctness guarantees. The Total Variation (TV) distance is used to quantify the distance between distributions, allowing for efficient computation and optimization of parameters. The effectiveness of the approach is demonstrated on benchmarks from the control community. The content is structured into sections discussing introduction, problem formulation, system description, total variation bounds, convergence analysis, algorithm details, experimental results, and proofs.
Introduction:
- Uncertainty propagation in complex autonomous systems.
- Importance of considering non-linear stochastic dynamics for safety-critical applications.
Problem Formulation:
- Need for efficient frameworks for uncertainty propagation.
- Various methods proposed but lacking error bounds on approximation error.
System Description:
- Discrete-time stochastic process representation.
- Transition kernel definition and probability distribution description.
Total Variation Bounds:
- Definition of Total Variation (TV) distance.
- Use of TV distance to quantify closeness between distributions.
Convergence Analysis:
- Proof showing approximation error can be minimized by increasing mixture size.
Algorithm Details:
- Iterative process based on mixture distribution approximation.
- Refinement procedure to optimize grid partitions for accuracy.
Experimental Results:
- Evaluation on benchmarks including linear systems and Dubins car model.
- Comparison with standard approaches showing improved TV bounds.
Proofs:
- Detailed proofs provided for Theorem 1, Corollary 1, and Theorem 2.
Citações
"Modern autonomous systems are becoming increasingly complex."
"Various methods have been proposed to propagate uncertainty in non-linear stochastic systems."