The document discusses various methods for analyzing identifiability and observability in nonlinear state and parameter estimation models, which are important concepts in system identification. It describes methods like local sensitivity analysis, empirical observability Gramians, asymptotic analysis, and the Alternating Conditional Expectation algorithm for testing structural and practical identifiability. It also discusses differential algebra approaches for testing observability in nonlinear systems.

1. An overview of Methods for analysis of Identifiability and Observability in Nonlinear State and Parameter Estimation Steinar M. Elgsæter Trial Lecture - October 14 2008

4. What is Identifiability and Observability?

5. Example: An oil well Reservoir Oil,water and gas rates Choke Well Pipeline Pressures and temperature Density Choke opening Unmeasured Measured

6. Internal system dynamics un-modeled disturbances output(y) input(u) Map internal states (x) System

7. An open-loop ”ballistic” state estimator internal states (x) modeled internal states (x) Internal dynamics measured output (y) Map Modeled internal dynamics modeled output(y) input(u) Map Plant Model Fitted parameters ( θ )

8. Closing the loop improves estimates in the face of uncertainty or disturbances Plant Model State Estimator - input output state injection term fitted state Plant Model Parameter Estimator - input output parameter injection term fitted parameter Duality Feedback loop

13. Flavors of nonlinear filters Type Constrained Solution Model considered Linearized Kalman Filter Unconstrained Explicit 1.Order local approximation Extended Kalman Filter Unconstrained Explicit 1. Order local approximation Unscented Kalman Filter Unconstrained Explicit 2.Order local approximation Moving Horizon Estimator Constrained Numerical Full model

16. Methods for testing identifiability

18. Structural identifiability

23. ” Practical” identifiability

28. Methods for testing observability* * and identifiability as a special case

36. Discussion/Conclusions

40. Thank you for your attention