Modelling and Control of Nonlinear Dynamic Systems with Gaussian Process Models

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Modelling and Control of Nonlinear Dynamic Systems with Gaussian Process Models. Juš Kocijan 1,2. 1 Jožef Stefan Institute, Ljubljana 2 Nova Gorica Polytechnic, Nova Gorica. Introduction. The method takes roots from statistics (Bayesian approach)

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### Modelling and Control of Nonlinear DynamicSystems with Gaussian Process Models

Juš Kocijan1,2

1 Jožef Stefan Institute, Ljubljana

2 Nova Gorica Polytechnic, Nova Gorica

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

CAPE Forum 2004, Veszprem, February 2004

GP Principle

The left plot shows Gaussian prediction at new point x1, conditioned on the training points (dots) while the right plot shows predictive mean along with its 2s error bars for two points, x2 that is close to training ones and x1 that is more distant

CAPE Forum 2004, Veszprem, February 2004

Dynamical systems identification with GP
• Dynamical systems: MAC project - EU 5th framework RTN
• From statical nonlinearities to dynamic systems: the same approach as for ANN or fuzzy models
• Difference: propagation of uncertainty

CAPE Forum 2004, Veszprem, February 2004

Ilustrative example:
• 1st order nonlinear dynamic system:
• 1st order model
• Function f is a GP (twodimensional regression model,

D = 2)

• Hyperparameters: v0, v1, w1, w2

CAPE Forum 2004, Veszprem, February 2004

Validation response

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)

CAPE Forum 2004, Veszprem, February 2004

Process control example – CSTR model validation

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])

CAPE Forum 2004, Veszprem, February 2004

TANH process: control results – constrained case (constraint on variance only)

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

CAPE Forum 2004, Veszprem, February 2004