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Mining Hybrid Models from Data Claire J Tomlin Hamsa Balakrishnan Inseok Hwang Department of Aeronautics and Astronautics Department of Electrical Engineering Stanford University May 2005 Background and Outline Tools for the analysis and control of hybrid systems Reachable set calculations Approximation algorithms for trajectory optimization in hybrid systems This talk identifying hybrid models from data 1 ETMS UAV flight data 3200 3000 3400 3500 3600 4000 3800 4500 5000 4000 3200 4200 4400 3300 5500 3400 6000 3500 4600 4400 4600 6500 4800 3600 x 5000 in m 5200 5400 5600 7000 3700 7500 3000 3800 4000 5000 6000 7000 8000 9000 3900 4000 4100 4200 4300 4800 5000 5200 5400 5600 5800 6000 UAV data with Teo and Jang ETMS data from McNally Ames ETMS data with synthesized flight with Alex Bayen Pascal Grieder 2 Drosophila wing epithelium Dsh protein distal Courtesy Dali Ma Stanford University Rousset et al Genes Dev 15 658 71 2001 Dsh p d Dsh a p proximal Time hrs Amonlirdviman Khare Tree Chen Axelrod Tomlin Science 05 Hybrid System Model 3 Stochastic Linear Hybrid System Mode Continuous state parameters Transition matrix Assumptions on system behavior We assume that Measurement matrices C are known System has a minimum known dwell time Td in each mode Typical system behavior is manifested in the available data sets Mode transitions are independent of the continuous state Mode transitions are probabilistic and Markovian 4 Maximum Likelihood Hybrid Model s1 1 l1 2 s2 l2 1 s3 l3 2 sNs lNs 1 t T Data sequence discrete modes segments Switching points Labels modes Continuous model Discrete model Maximum Likelihood Model Given the continuous output of the system we would like to compute the maximum likelihood hybrid system model where is the likelihood function we would like to maximize 5 Parameter Inference Algorithm Assume an initial continuous model 0 and an initial discrete model D 0 iterate Step 1 Find the globally optimal segmentation points S and their labels L assuming

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