# Statistical Results on Filtering and Epi-convergence for Learning-Based Model Predictive Control

### Anil Aswani, Humberto Gonzalez, S. Shankar Sastry and Claire Tomlin

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2011-153

December 17, 2011

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2011/EECS-2011-153.pdf

Learning-based model predictive control (LBMPC) is a technique that provides deterministic guarantees on robustness, while statistical identification tools are used to identify richer models of the system in order to improve performance. This technical note provides a result that elucidates the reasons for the choice of measurement model used with LBMPC, and it gives proofs concerning the stochastic convergence of LBMPC. The first part of this note discusses simultaneous state estimation and statistical identification (or learning) of unmodeled dynamics, for dynamical systems that can be described by ordinary differential equations (ODE's). The second part provides proofs concerning the epi-convergence of different statistical estimators that can be used with the LBMPC technique. In particular, we prove results on the statistical properties of a nonparametric estimator that we have designed to have the correct deterministic and stochastic properties for numerical implementation when used in conjunction with LBMPC.

BibTeX citation:

@techreport{Aswani:EECS-2011-153, Author = {Aswani, Anil and Gonzalez, Humberto and Sastry, S. Shankar and Tomlin, Claire}, Title = {Statistical Results on Filtering and Epi-convergence for Learning-Based Model Predictive Control}, Institution = {EECS Department, University of California, Berkeley}, Year = {2011}, Month = {Dec}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2011/EECS-2011-153.html}, Number = {UCB/EECS-2011-153}, Abstract = {Learning-based model predictive control (LBMPC) is a technique that provides deterministic guarantees on robustness, while statistical identification tools are used to identify richer models of the system in order to improve performance. This technical note provides a result that elucidates the reasons for the choice of measurement model used with LBMPC, and it gives proofs concerning the stochastic convergence of LBMPC. The first part of this note discusses simultaneous state estimation and statistical identification (or learning) of unmodeled dynamics, for dynamical systems that can be described by ordinary differential equations (ODE's). The second part provides proofs concerning the epi-convergence of different statistical estimators that can be used with the LBMPC technique. In particular, we prove results on the statistical properties of a nonparametric estimator that we have designed to have the correct deterministic and stochastic properties for numerical implementation when used in conjunction with LBMPC.} }

EndNote citation:

%0 Report %A Aswani, Anil %A Gonzalez, Humberto %A Sastry, S. Shankar %A Tomlin, Claire %T Statistical Results on Filtering and Epi-convergence for Learning-Based Model Predictive Control %I EECS Department, University of California, Berkeley %D 2011 %8 December 17 %@ UCB/EECS-2011-153 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2011/EECS-2011-153.html %F Aswani:EECS-2011-153