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