Dynamic Systems Identification with Gaussian Process Models

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Dynamic Systems Identification with Gaussian
Process Models

• Juš Kocijan
• Jožef Stefan Institute, Ljubljana, Slovenia
• University of Nova Gorica, Nova Gorica, Slovenia

Seminar Ceské Spolecnosti pro Kybernetiku a
Informatiku , October 2009, Prague, Czech republic
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Motivation

• Topic on overview of dynamic systems
  identification with Gaussian process models (GP
  models)
• Problem application of machine learning approach
  for dynamic systems modelling and its
  applications
• Theoretical solution conventional approach
  delayed input and output values as regressors
• Validation of theory applications in various
  domains since 1999

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Identification why and how

• Theoretical modelling first principles
  modelling vs. identification
• Dynamic system identification ? model ? e.g.
  prediction, automatic control, ...
• Nonlinear dynamic system identification
• problems ? ANN, fuzzy models, ...
• difficult to use (structure determination, large
• number of parameters, lots of training data)
• the issue of confidence in model
• GP model recent complementary approach

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Historical overview

• GP literature in the field of statistics, where
  this approach originates
• Kriging in the geophysics literature
• GP in curve fitting and regression OHagan, 1978
  (triggered no special attention)
• Relation with ANN Neal, 1996
• Further developments of GP regression Williams
  and Rasmussen, 1996 (modelling of static
  non-linearities)
• use of GP for dynamic systems EU 5th framework
  RTN - MAC project (2000-2004)
• GP models, GP priors, GP regression, GP Dynamical
  Models

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http//dsc.ijs.si/jus.kocijan/GPdyn/
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GP model

• Probabilistic (Bayes) model.
• Nonparametric model no predetermined structure
  (basis functions) depending on system
• Determined by
• Input/output data (data points, not
  signals)
• (learning data identification data)
• Covariance matrix

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Gaussian process model
Bayes based modelling
y
x
GP model
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GP model

• Prediction of the output based on similarity
  test input training inputs
• Output normal distribution
• Predicted mean
• Prediction variance

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Covariance function

• Covariance function
• functional part and noise part
• stationary/unstationary, periodic/nonperiodic,
  etc.
• Expreses prior knowledge about system properties,
  
• frequently Gaussian covariance function
• Smooth function
• Stationary function

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Hyperparameters

• Identification of GP model optimisation of
  covariance function parameters
• Optimisation
• Cost function maximum likelihood of data for
  learning

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Static illustrative example

• Static example
• 9 learning points
• Grey band
• Rare data density ?
• increased variance
• (higher uncertainty).

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Dynamic systems

• Static vs. Dynamic
• Dynamic models
• conventional approach (ANN, fuzzy models,
  etc.) is
• delayed inputs and outputs as regressors
• Input/output training pairs xi/yi
• xi ... regresor values (GPARX model)
• u(k-1),..,u(k-L),
  y(k-1),..,y(k-L)
• yi ... system output values
• y(k)

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Dynamic system identification and model simulation

• Why does identification of dynamic systems seem
  more complex than modelling of static functions?
• Simulation
• naive ... m(k)
• with propagation
• m(k),v(k)
• Analytic app.
• Taylor app.
• exact
• MC Monte Carlo
• with mixtures

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GP model attributes (vs. e.g. ANN)

• Smaller number of parameters
• Measure of confidence in prediction, depending on
  data
• Incorporation of prior knowledge
• Easy to use (practice)
• Computational cost increases with amount of data
  ?
• Recent method, still in development
• Nonparametrical model

(also possible in some other models)
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Identification challenges

• Methodology of experimental modelling for dynamic
  systems based on GP models
• Procedure, suggestions, examples, etc.
• Incorporation of prior knowledge
• Nonparametric model
• ? Utility of the method

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Identification case study Bioreactor

• Identification procedure, properties of obtained
  GP model ?
• Bioreactor discrete nonlinear 2nd order dynamic
  system

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Bioreactor (2)

• Defining the purpose of the model response
  prediction
• Model selection
• Gaussian covariance function
• (stationarity, smoothness)
• Regressors selection
• Design of the experiment
• Input/output signal ?
• 600 data for identification
• Realisation of the experiment,
• data processing

Input and output signals used for generating data
for identification
y(k-1)y(k-L) u(k-1)u(k-L)
y(k-1)y(k-L) u(k-1)u(k-L), y(k)
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Bioreactor (3)

• Model training
• Optimisation of hyperparameters
• Model validation
• plausibility (looks, behaves logical)
• falseness (I/O inspection)
• purposiveness (satisfaction of the purpose)

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Bioreactor (4)

• Model validation
• plausibility
• qualitative (visualy from I/O response)
• quantitative cost functions
• Mean squared error (MSE),
• Mean relative square error (MRSE),
• Log predictive density error (LD),
• Negative log-likelihood of the training data
  (LL).

variance
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Bioreactor (5)

• validation ? L2
• ARD ? the number of regressors is reduced

u(k-1) u(k-2) y(k-1) y(k-2)
u(k-1) u(k-2) y(k-2)
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Bioreactor (5)

• Simulation in the trained region, but not with
  the identification signal

Simulated response with 95 confidence band and
error
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Bioreactor (6)

• Simulation in the not modelled region
• u(k) gt 0.7.

Simulation result in the not modelled region
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Bioreactor (7)

• More noise
• s 2(ymax- ymin)
• Increased variance
• Increased noise
• Insufficient data

Simulation result with more noise in
identification signal, s0.002
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Bioreactor (8)

• Unmodelled input
• z0.05 for tgt30s
• ?
• Prediction confidence is not changed in the
  modelled region

Simulation result in the case of not modelled
input, z0.05 for tgt30 s
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Applications and domains of use

• dynamic systems modelling
• time-series prediction
• dynamic systems control
• fault detection
• smoothing
• chemical engineering and process control
• biomedical engineering
• biological systems
• environmental system
• power systems and engineering
• motion recognition
• traffic

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Incorporation of local linear models (LMGP model)

• Derivative of function observed beside the values
  of function
• Derivatives are coefficients of linear local
  model in an equilibrium point (prior knowledge)
• Covariance function to be replaced the procedure
  equals as with usual GP
• Very suited to data distribution that can be
  found in practice

J. Kocijan and A. Girard. Incorporating linear
local models in Gaussian process model. In
Proceedings of IFAC 16th World Congress, Praga,
2005.
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Applications for control design

• General model based predictive control principle
• Cost function (PFC)
• constraints on input signal, input signal rate,
  state signals, state signals rate and
• constrained optimisation SAFE CONTROL

J. Kocijan and R. Murray-Smith. Nonlinear
predictive control with Gaussian process
model. In Switching and Learning in Feedback
Systems, volume 3355 of Lecture Notes in Computer
Science, Pages 185-200. Springer, Heidelberg,
2005. B. Likar and J. Kocijan. Predictive
control of a gas-liquid separation plant based on
a Gaussian process model. Computers and Chemical
Engineering, Volume 31, Issue 3, Pages 142-152,
2007.
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pH process control results constrained case
(constraint on variance only)
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pH process control results constrained case
(constraint on variance only)
Next step Explicit Nonlinear Predictive Control
Based on Gaussian Process Models
A. Grancharova, J. Kocijan and T. A. Johansen.
Explicit stochastic predictive control of
combustion plants based on Gaussian process
models. Automatica, Volume 44, Issue 6, Pages
1621-1631, 2008.
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Application of GP models for fault diagnosis and
detection

• Is the fault diagnosed because of the fault
  occurance or because model is not OK?

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Dj. Juricic and J. Kocijan. Fault detection
based on Gaussian process model. In I. Troch and
F. Breitenecker, editors, Proceedings of the 5th
Vienna Symposium on Mathematical Modeling
(MathMod), Vienna, 2006.
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Conclusions

• The Gaussian process model is an example of a
  flexible, probabilistic, nonparametric model with
  inherent uncertainty prediction
• It is suitable for dynamic systems modelling
• When to use GP model?
• systems nonlinearity, corrupted data (noise,
  uneven distribution),
• insufficient prior knowledge, uncertainty
• biological, environmental systems, etc.
• A case study for the illustration of
  identification procedure.

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Conclusions

• The research of GP modells for dynamic systems is
  growing.
• Further work
• Software
• Analytical tools
• Application niches