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
Hybrid System Model
• Complex, multi-modal systems• Can combine probabilistic, discrete techniques with control of
continuous systems
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
Online System Identification
Online System Identification
Online System Identification
[Bickel and Li, 2007]
• Undersampling for high-dimensional systems• Constrained dynamics• Fast-slow dynamics
Online System Identification
Online System Identification
Online System Identification
Online System Identification
Look for a geometric structure for sparsityLocal linear (hybrid) models are easy to manipulate
Online System Identification
Local Linear Regression
Solve for in for all
Rewrite as:
where
•Difficulty in interpreting regression coefficients•Gradient of function does not exist
@f@x1
= limh! 0
f (x1 + h;x2) ¡ f (x1;x2)h
Online System Identification
Exterior derivative of function does exist• Extension of gradients to manifolds• Best local linear approximation of function on manifold
df = A : limkhk! 0
x+h2M
kf (x + h) ¡ f (x) ¡ Ahkkhk
= 0
Online System Identification
1515(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