Uncertainty Propagation in Stochastic Systems via Mixture Models with Error Quantification (2403.15626v2)
Abstract: Uncertainty propagation in non-linear dynamical systems has become a key problem in various fields including control theory and machine learning. In this work we focus on discrete-time non-linear stochastic dynamical systems. We present a novel approach to approximate the distribution of the system over a given finite time horizon with a mixture of distributions. The key novelty of our approach is that it not only provides tractable approximations for the distribution of a non-linear stochastic system, but also comes with formal guarantees of correctness. In particular, we consider the total variation (TV) distance to quantify the distance between two distributions and derive an upper bound on the TV between the distribution of the original system and the approximating mixture distribution derived with our framework. We show that in various cases of interest, including in the case of Gaussian noise, the resulting bound can be efficiently computed in closed form. This allows us to quantify the correctness of the approximation and to optimize the parameters of the resulting mixture distribution to minimize such distance. The effectiveness of our approach is illustrated on several benchmarks from the control community.
- D. Landgraf, A. Völz, F. Berkel, K. Schmidt, T. Specker, and K. Graichen, “Probabilistic prediction methods for nonlinear systems with application to stochastic model predictive control,” Annual Reviews in Control, vol. 56, p. 100905, 2023.
- È. Pairet, J. D. Hernández, M. Carreras, Y. Petillot, and M. Lahijanian, “Online mapping and motion planning under uncertainty for safe navigation in unknown environments,” IEEE Transactions on Automation Science and Engineering, vol. 19, no. 4, pp. 3356–3378, 2021.
- A. Girard, C. Rasmussen, J. Q. Candela, and R. Murray-Smith, “Gaussian process priors with uncertain inputs application to multiple-step ahead time series forecasting,” Advances in neural information processing systems, vol. 15, 2002.
- T. S. Schei, “A finite-difference method for linearization in nonlinear estimation algorithms,” Automatica, vol. 33, no. 11, pp. 2053–2058, 1997.
- J. Duník, S. K. Biswas, A. G. Dempster, T. Pany, and P. Closas, “State estimation methods in navigation: Overview and application,” IEEE Aerospace and Electronic Systems Magazine, vol. 35, no. 12, pp. 16–31, 2020.
- K. Ito and K. Xiong, “Gaussian filters for nonlinear filtering problems,” IEEE transactions on automatic control, vol. 45, no. 5, pp. 910–927, 2000.
- J. S. Lui and R. Chen, “Sequential monte carlo methods for dynamic systems,” Journal of the American Statistical Association, vol. 93, no. 443, p. 1032, 1998.
- M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-gaussian bayesian tracking,” IEEE Transactions on signal processing, vol. 50, no. 2, pp. 174–188, 2002.
- K. Polymenakos, L. Laurenti, A. Patane, J.-P. Calliess, L. Cardelli, M. Kwiatkowska, A. Abate, and S. Roberts, “Safety guarantees for iterative predictions with gaussian processes,” in 2020 59th IEEE Conference on Decision and Control (CDC). IEEE, 2020, pp. 3187–3193.
- A. Jasour, A. Wang, and B. C. Williams, “Moment-based exact uncertainty propagation through nonlinear stochastic autonomous systems,” arXiv preprint arXiv:2101.12490, 2021.
- F. Weissel, M. F. Huber, and U. D. Hanebeck, “Stochastic nonlinear model predictive control based on gaussian mixture approximations,” in Informatics in Control, Automation and Robotics: Selected Papers from the International Conference on Informatics in Control, Automation and Robotics 2007. Springer, 2009, pp. 239–252.
- G. Terejanu, P. Singla, T. Singh, and P. D. Scott, “Uncertainty propagation for nonlinear dynamic systems using gaussian mixture models,” Journal of guidance, control, and dynamics, vol. 31, no. 6, pp. 1623–1633, 2008.
- A. L. Gibbs and F. E. Su, “On choosing and bounding probability metrics,” International statistical review, vol. 70, no. 3, pp. 419–435, 2002.
- L. Devroye, A. Mehrabian, and T. Reddad, “The total variation distance between high-dimensional gaussians with the same mean,” arXiv preprint arXiv:1810.08693, 2018.
- A. B. Tsybakov, “Introduction to nonparametric estimation,” Springer Series in Statistics, 2009.
- F. B. Mathiesen, S. C. Calvert, and L. Laurenti, “Safety certification for stochastic systems via neural barrier functions,” IEEE Control Systems Letters, vol. 7, pp. 973–978, 2022.
- M. Šimandl, J. Královec, and T. Söderström, “Advanced point-mass method for nonlinear state estimation,” Automatica, vol. 42, no. 7, pp. 1133–1145, 2006.
- S. Adams, M. Lahijanian, and L. Laurenti, “Formal control synthesis for stochastic neural network dynamic models,” IEEE Control Systems Letters, vol. 6, pp. 2858–2863, 2022.
- T. Brüdigam, M. Olbrich, D. Wollherr, and M. Leibold, “Stochastic model predictive control with a safety guarantee for automated driving,” IEEE Transactions on Intelligent Vehicles, vol. 8, no. 1, pp. 22–36, 2021.
- Q. H. Ho, Z. N. Sunberg, and M. Lahijanian, “Gaussian belief trees for chance constrained asymptotically optimal motion planning,” in 2022 International Conference on Robotics and Automation (ICRA). IEEE, 2022, pp. 11 029–11 035.
- S. Barsov and V. V. Ulyanov, “Estimates of the proximity of gaussian measures,” in Sov. Math., Dokl, vol. 34, 1987, pp. 462–466.
- B. K. Driver, “Math 280 (probability theory) lecture notes,” Lecture notes, University of California, San Diego, Department of Mathematics, vol. 112, pp. 92 093–0112, 2007.