1. Hybrid Systems: Model Identification and State Estimation Hamsa Balakrishnan, David Culler, Edward A. Lee, S. Shankar Sastry, Claire Tomlin (PI) University of California at Berkeley December 17 2009

2. Hybrid System Model Complex, multi-modal systems Can combine probabilistic, discrete techniques with control of continuous systems

3. Some results… Model ID: for stochastic linear hybrid systems, with mode switching governed by a Markovian switching matrix –Iteratively maximizing the likelihood of the discrete model and then finding the maximum likelihood continuous model [Balakrishnan et al, 2004] State estimation: –both discrete and continuous [Hwang, Balakrishnan et al, 2003] –asynchronous

4. Online System Identification

7. [Bickel and Li, 2007] Undersampling for high-dimensional systems Constrained dynamics Fast-slow dynamics Online System Identification

11. Look for a geometric structure for sparsity Local linear (hybrid) models are easy to manipulate Online System Identification

12. Local Linear Regression Solve for in for all Rewrite as: where

13. Difficulty in interpreting regression coefficients Gradient of function does not exist Online System Identification

14. Exterior derivative of function does exist Extension of gradients to manifolds Best local linear approximation of function on manifold Online System Identification

15. 15 (Aswani et al., submitted 2009); (Bickel and Levina, 2008) Locally learn manifold Constrain regression vector to lie on the manifold by penalizing for deviations from manifold Where is chosen to penalize for lying off of the manifold New Estimation Approach