1. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw 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

2. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw 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

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

4. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw 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

5. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw p(y) * * * * y x x0x0 * | x=x 0 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:

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

7. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw 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)

8. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw y = f(x) = = cos (6x 2 ) Staticexample GP model x y

9. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw GP model Dynamic system Input/output training pairs x i /y i x i... regressor values [u t-1,..,u t-k, y t-1,..,y t-k ] y i... system output y t Simulation “naive”... m(k)

10. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw Problem of nonlinear dynamic systems identification Engine example – longitudinal dynamics

11. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw 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

12. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw

13. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw Case study: gas-liquid separator

14. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw Nonlinearity of the system Model structure:

15. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw Model identification Seven equilibrium points Seven linear LM (14 points) 60 off equilibrium points

16. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw Model validation SE=0.00056 LD=-1.97

17. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw Model validation

18. Jožef Stefan Institute Department of Systems and Control Systems Science XVI, September 2007, Wroclaw 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.