Modelling and Control of Nonlinear Dynamic Systems with Gaussian Process Models

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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
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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

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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
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Dynamical systems identification with GP

• Dynamical systems MAC 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

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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

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Validation response
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Uncertainty surface
Uncertainty surface (left plot)
?(k1)f(u(k),y(k)) for the GP approximation and
location of training data (right plot)
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Process control example CSTR model validation
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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,hyperpara
  meters,inverse covariance matrix)

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