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An overview of Methods for analysis of Identifiability and Observability in Nonlinear State and Parameter Estimation

Steinar M. ElgsæterTrial Lecture - October 14 2008

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Emphasis of this presentation

• identifiability and observability in the context of industrial processes, where even simplified models may have 30 or more states.

• emphasis on recent advances in computational methods (many motivated by applications in systems biology)

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Presentation overview

• what is identifiability and observability?• methods for testing identifiability

– structural identifiability

– ”practical” identifiability

• methods for testing observability• discussion/conclusions

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What is Identifiability and Observability?

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Example: An oil well

Reservoir

Oil,water and gas rates

Choke

Well

Pipeline

Pressures and temperature

Density

Chokeopening

Unmeasured

Measured

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Internal system

dynamics

un-modeled disturbances

output(y)input(u)Map

internal states (x)

System

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internal states (x)

modeled internal states (x)

An open-loop ”ballistic” state estimator

Internal dynamics

measured output (y)Map

Modeled internal

dynamics

modeled output(y)

input(u)

Map

Plant

Model

Fitted parameters (θ)

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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

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Parameter estimation and identifiability• ”Whether the identification procedure will yield a

unique value of the parameter θ, and/or whether the resulting model is equal to the true system.”1

• identifiability both a property of the model structure (1) and the data set (2) 1 :

1. no two parameter values should give the same model

2. the data should be informative enough to distinguish between non equal models

1. Ljung,(1999) System Identification: Theory for the user

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State estimation and observability• observability

– loosely: ”is it possible to determine the initial state of a system by observing its input and output over some period”?

– different definitions for nonlinear systems depending on the theoretical approach chosen

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Observability in linear systems

• definition: A linear system is observable if any unknown initial state can be determined uniquely by observing inputs and outputs over some time interval

• observability can be tested for linear systems using linear algebra (for instance by the rank of the observability matrix or Gramian)

• note that observability does not depend on trajectory for linear systems

1. Ljung,(1999) System Identification: Theory for the user

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Kalman Filter• often used to simultaneously estimate both parameters

and states • assumes white, zero-mean measurement and process

noise,– implies no structural mismatch

• requires linear state and measurement models• ad-hoc adjustment of noise covariances often required

when applied to complex, nonlinear chemical processes– divergence problems often cited(1 and references therein)

1. Wilson, D.I., Agarwal, M, Rippin, D.W.T “Experiences implementing the extended Kalman filter on an industrial batch reactor”(1998), Computers Chem Engng, Vol 22, No.11, pp.1653-1672

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Flavors of nonlinear filtersType 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

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Parameter estimation a special case of state estimation

• theoretically θ is a special kind of state where• the distinction between identifiability and observability

may be slightly artificial, – many methods discussed are extendable to both types of analysis

– parameters are estimated alongside states in Kalman filters in practice anyway...

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Why is observability and identifiability an issue?• less instruments than one would like

– extreme pressures, temperatures or inaccessibility

• overparametrized models• many simplified process models are still nonlinear

and have 30 states or more– while linear models can be easily both solved and analyzed

analytically, nonlinear process control models can usually not even be solved analytically

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Methods for testing identifiability

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Different methods for different aspects of identifiability

1. a priori ”structural identifiability” 1. sensitivity analysis

2. empirical observability Gramians

2. a posteriori ”practical identifiability” including data1. asymptotic analysis (local)

2. Alternating Conditional Expectation Algorithm

3. algebraic analysis

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Structural identifiability

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Local Sensitivity Analysis1

• which parameters can be uniquely fitted against data?

• sensitivity matrix:

– can rank parameters to find sets which have high sensitivity and low collinearity

– can be assessed locally and numerically for a range of (u,θ) for nonlinear systems

1. Lund, B.F. (2005) ”Rigorous simulation models for improved process operation” (PhD thesis)

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Local Sensitivity Analysis(2)• advantage:

– based on intuitive principles

– sensitivity matrices are calculated quickly and easily

• disadvantages:– physical insight required to choose candidate operating points (u,θ)

– interpreting sensitivity matrices may get technical

– an iterative approach may be required to find a suitable parameter set

– linear and local analysis of nonlinear systems

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Empirical Observability Gramians(1)• a linear stable system is observable if the observability

Gramian, an energy function, is of full rank • applying energy functions directly to nonlinear systems

is computationally intensive• if nonlinear systems could be linearized, to determine

linear observability Gramians, but significant dynamic effects could be neglected

• empirical Gramians originally introduced for model reduction1

1. Singh, A. K. and Hahn, J. On the use of empirical gramians for controllability and observability analysis. In Proc. 2005 American Control Conference (ACC). Portland,OR, USA, 2005 pp. 140-141

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Empirical Observability Gramians(2)1

• approximation of analytical Gramian, valid for a region

• reduces to linear Gramian for linear systems• both observability and identifiability2

1. J. Hahn, T.F. Edgar, (2002)“An improved method for nonlinear model reduction using balancing of empirical gramians”, Computers and Chemical Engineering, Vol. 26, 2002, pp. 1379-1397.

2. Geffen, D. (2008) Parameter Identifiability of Biochemcial Reaction Networks in Systems Biolgy (Master Thesis) , Queens University, Ontario

”Regional” approximation found from simulated perturbations of nonlinear model

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”Practical” identifiability

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Asymptotic analysis1

• distribution of covariance for– asymptotic analysis (large N)

– covariance matrix found from a second order Taylor approximation of the model around an specific parameter value

– quality tag or confidence intervals for fitted parameters

– estimated from data

– for linear systems with invertible covariance matrix:

1. Ljung,(1999) System Identification: Theory for the user

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Shortcomings of asymptotic analysis• N is never infinite, it is finite and often small• for nonlinear models, analysis is only local

– which operating point(s) to analyze?

• calculations require inverting matrices which are close to singular if information content is low, – covariance estimates may be inaccurate when covariance is large

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Parameter uncertainty estimates can assess how close to identifiability a model/dataset is

• covariance-matrix analysis• Leave-Out-Sign-Dominant Correlation Regions1

• brute-force computational methods (bootstrapping2)

1. Campi, M.C. and Weyer, E. (2005) ”Guaranteed non-asymptotic confidence regions in system identification”, Automatica, 41(10), 1751-1764

2. Efron, B. and Tibshirani, R. J. (1993) An Introduction to the Bootstrap, Chapmann & Hall

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Alternating Conditional Expectation Algorithm1

• principle: non-identifiabilities in nonlinear dynamical model reveal themselves in functional dependencies between parameters– model fitted repeatedly in bootstrapping to

data to generate a set of fitted parameters

– functional dependencies in this set are identified using ACE methods2

1. S. Hengl, C. Kreutz, J. Timmer and T. Maiwald (2007) Data-based identifiability analysis of non-linear dynamical models,Bioinformatics, Vol. 23 no. 19, pp 2612–2618

2. Breiman,L. and Friedman,J. (1985) Estimating optimal transformations for multiple regression and correlation. J. Am. Stat. Assoc., 80, 580–598.

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Methods for testing observability*

* and identifiability as a special case

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Observability tests in nonlinear systems• directions:

– differential geometry: Indistinguishable Dynamics rank condition (not considered)

– differential algebra: algebraic observability rank condition

• note: observability of nonlinear systems will depend on trajectory in general

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Differential algebra: algebraic observability• ”The state variable xi is observable if there exists an

algebraic relation that binds xi to inputs, outputs and a finite number of their derivatives.”1

• Example2

1. Diop, S. and Fliess, M. On nonlinear observability In Proceedings of the first european control conference (Grenoble, France, July 2-5 1991), C.Commault and coll., vol. 1., Hermès, pp.152-157

2. Anguelova 2004, Nonlinear Observability and Identifiability: General Theory and a Case Study of a Kinetic Model for S. cerevisiae (Master Thesis), Chalmers, Göteborg University

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Differential algebra: algebraic observability• the algebraic observability of the

system

Σ requires the full rank of matrix,

where n is the number of states.1

• possible to pinpoint unobservable states by removing columns of matrix and re-calculating rank iteratively

1. Diop, S. and Wang.Y (1993) ”Equivalence between algebraic observability and local generic observability ”, in Proc. 32nd Conference on Decision and Control

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Local Algebraic Observability (3)...• for bilinear models, if more than two derivations are

required express a model in terms of derivates of y and u, numerical solution may be required (3.order polynomials)

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Computational algebraic observability analysis• symbolic calculations of Lie-derivatives not possible in

polynomial time– instead ”semi-numerical” algorithm, implemented in Maple1

– applies only to rational control systems– specializes parameters to random integer values and inputs to power series of

t with integer coefficients– achieves polynomial run-time

• algorithm output– a positive observability test is always correct– a negative observability test is correct with a high probability– non-observable states and parameter symmetries determined in exponential

time

1. Sedoglavic A. 2002, A probabilistic algorithm to test local algebraic observability in polynomial time, Journal of Symbolic Computation 33:735-755

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Computational algebraic observability analysis(2)• only one application of CAOA found in

literature1

– case study of kinetic model for metabolic dynamics in biochemistry

• Sedoglavic algorithm applied:– 21 states, all states measured, ~100 parameters– 2 non-identifiable parameters identified– two ”symmetries” identified, including 11

parameters, – one parameter in each symmetry must be fixed

to make model identifiable

1. Anguelova 2004, Nonlinear Observability and Identifiability: General Theory and a Case Study of a Kinetic Model for S. cerevisiae (Master Thesis), Chalmers, Göteborg University

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Determining observability robustness through simulations1

• observability Jacobian determined for different initial conditions and different parameter values in simulations

• did not relate findings to individual states

1. Dafis C.J, Nwankpa, C.O. (2005) Characteristics of Degree of Observability Measure for Nonlinear Power Systems, Proc. 38th Hawaii Int. Conf. on System Sciences

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Discussion/Conclusions

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”Observability in practice”

• observability is currently treated as– an ”on-off” property’

– given perfect measurements

• in practice states are not equally observable.• it would be useful to develop a ”measure of

observability” for each individual state which assesses for instance – sensitivity to non-idealness of measurements (noise)

– sensitivity to operating point/parameter value

– sensitivity to excitation

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Conclusion

• several recent advances in computational methods for observability analysis have been made:– empirical Observability Gramian

– Sedoglavic algorithm for algebraic observability

– Alternating Conditional Expectation algorithm

• these methods appear well-suited to process control – computing time is ”cheap”

– ”turnkey methods”: computational methods are not model-specific, applying methods in practice need to be laborious

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Conclusion(2)

• application of computational observability analysis in process control– no reference of such applications found in literature

– advances in field mean that timing may be good for a case study in process control

– a research topic well-suited to industry-university collaboration (Master thesis topic?)

– mainly of interest for industry which has design of nonlinear observers for process control as core business (narrow)

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Thank you for your attention