1. “Jožef Stefan” Institute Department of Systems and Control Modelling and Control of Nonlinear Dynamic Systems with Gaussian Process Models Juš Kocijan Juš Kocijan 1,2 1 Jožef Stefan Institute, Ljubljana 2 Nova Gorica Polytechnic, Nova Gorica

2. “Jožef Stefan” Institute Department of Systems and Control CAPE Forum 2004, Veszprem, February 2004 Introduction The method takes roots from statistics (Bayesian approach) A Gaussian process (GP) is a collection of random variables which have a joint multivariate Gaussian distribution Use of stochastic variables (vectors) Relatively sophisticated theoretical background - relatively simple use Increasingly used for applications of Neural Networks Main questions: Modelling for control? When & why? Probabilistic nonparametric approach to modelling of dynamic systems

3. “Jožef Stefan” Institute Department of Systems and Control CAPE Forum 2004, Veszprem, February 2004 GP Principle The left plot shows Gaussian prediction at new point x 1, conditioned on the training points (dots) while the right plot shows predictive mean along with its 2  error bars for two points, x 2 that is close to training ones and x 1 that is more distant

4. “Jožef Stefan” Institute Department of Systems and Control CAPE Forum 2004, Veszprem, February 2004 Dynamical systems identification with GP Dynamical systems: MAC project - EU 5th framework RTNMAC project - EU 5th framework RTN Multi-step-ahead predictions From statical nonlinearities to dynamic systems: the same approach as for ANN or fuzzy models Difference: propagation of uncertainty

5. “Jožef Stefan” Institute Department of Systems and Control CAPE Forum 2004, Veszprem, February 2004 Ilustrative example: 1 st order nonlinear dynamic system: 1 st order model Function f is a GP (twodimensional regression model, D = 2) Hyperparameters: v 0, v 1, w 1, w 2

6. “Jožef Stefan” Institute Department of Systems and Control CAPE Forum 2004, Veszprem, February 2004 Validation response

7. “Jožef Stefan” Institute Department of Systems and Control CAPE Forum 2004, Veszprem, February 2004 Uncertainty surface Uncertainty surface (left plot)  (k+1)=f(u(k),y(k)) for the GP approximation and location of training data (right plot)

8. “Jožef Stefan” Institute Department of Systems and Control CAPE Forum 2004, Veszprem, February 2004 Process control example – CSTR model validation

9. “Jožef Stefan” Institute Department of Systems and Control CAPE Forum 2004, Veszprem, February 2004 Practical considerations on modelling of dynamic systems Nonparametric approaches are traditionally more popular in control engineering practice Variance that comes with model gives information about model validity in the region of use Computational issues – inverse of covariance matrix Documentation of the model (input,output,hyperparameters,[inverse covariance matrix])

10. “Jožef Stefan” Institute Department of Systems and Control CAPE Forum 2004, Veszprem, February 2004 TANH process: control results – constrained case (constraint on variance only)

11. “Jožef Stefan” Institute Department of Systems and Control CAPE Forum 2004, Veszprem, February 2004 Practical considerations on control with Gaussian processes Nonparametric model of nonlinear system: predictive control strategies NMPC not so popular due to difficulties to construct a model on a reliable and consistent basis Practical nonlinear robust control Computational load is a constraint Efforts are conducted to make process control application in industrial like environment