1 / 18

Gaussian Process Model Identification : a Process Engineering Case Study

Gaussian Process Model Identification : a Process Engineering Case Study. Juš Kocijan 1,2 , Kristjan Ažman 1 , 1 Jožef Stefan Institute, Ljubljana, Slovenia 2 University of Nova Gorica, Nova Gorica, Slovenia. Motivation:. Topic : nonlinear dynamic systems identification

dore
Télécharger la présentation

Gaussian Process Model Identification : a Process Engineering Case Study

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Gaussian Process ModelIdentification: a Process Engineering Case Study Juš Kocijan1,2, Kristjan Ažman1, 1Jožef Stefan Institute, Ljubljana, Slovenia 2University of Nova Gorica, Nova Gorica, Slovenia Systems Science XVI, September 2007, Wroclaw

  2. Motivation: • Topic: nonlinear dynamic systems identification • Problem: unballance between number of measurements in equilibrium and out of equilibrium • Theoretical solution:Gaussian process model with incorporated linear local modelsproblem solution + measure of confidence in prediction • Validation of theory: application in a process engineering case study Systems Science XVI, September 2007, Wroclaw

  3. Overview: • Modelling with Gaussian process (GP) priors • Incorporation of linear local models • Modelling case study of gas-liquid separator Systems Science XVI, September 2007, Wroclaw

  4. Identification – why and how • 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)  • GP model – reduces some of these problems Systems Science XVI, September 2007, Wroclaw

  5. y | p(y) x=x0 * * * * * x x0 GP model • Probabilistic, non-parametric model, constituted of: • covariance function • input/output data pairs (points, not signals) Prediction of the output based on similarity test input – training inputs Normally distributed output: Systems Science XVI, September 2007, Wroclaw

  6. Gaussian processes • Gaussian process – set of normally distributed random variables: • mean μ(X) • covariance matrix K(X) • Covariance function Gaussian • Optimisation: • cost function: log-density • method: maximum likelihood • optimisation: conjugate gradients Systems Science XVI, September 2007, Wroclaw

  7. 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) Systems Science XVI, September 2007, Wroclaw

  8. y x GP model Static example y = f(x) = = cos (6x2) Systems Science XVI, September 2007, Wroclaw

  9. GP model Dynamic system • Input/output training pairs xi/yi xi ... regressor values [ut-1,..,ut-k, yt-1,..,yt-k] yi ... system outputyt • Simulation • “naive” ... m(k) Systems Science XVI, September 2007, Wroclaw

  10. Problem of nonlinear dynamic systems identification Engine example – longitudinal dynamics Systems Science XVI, September 2007, Wroclaw

  11. 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 Systems Science XVI, September 2007, Wroclaw

  12. Systems Science XVI, September 2007, Wroclaw

  13. Case study: gas-liquid separator Systems Science XVI, September 2007, Wroclaw

  14. Nonlinearity of the system Model structure: Systems Science XVI, September 2007, Wroclaw

  15. Model identification • Seven equilibrium points • Seven linear LM (14 points) • 60 off equilibrium points Systems Science XVI, September 2007, Wroclaw

  16. Model validation SE=0.00056 LD=-1.97 Systems Science XVI, September 2007, Wroclaw

  17. Model validation Systems Science XVI, September 2007, Wroclaw

  18. Conclusions • The Gaussian process model is an example of a flexible, probabilistic, nonparametric model with inherent uncertainty prediction. • The GP model with incorporated local linear models (LMGP) is a possible solution for the problem of measurement data distribution in equilibrium and out of equilibrium. • The application of LMGP modelling method on a gas-liquid separator demonstrated feasibility of this solution in practice. Systems Science XVI, September 2007, Wroclaw

More Related