Constrained Parameter Estimation in Fuzzy Modeling

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Constrained Parameter Estimation in Fuzzy Modeling

• J. Abonyi, R. Babuka, M. Setnes, Verbruggen,
  F. Szeifert
• Control Engineering Laboratory
• Faculty of Information Technology and Systems
• Delft University of Technology
• Delft, the Netherlands
• e-mail J.Abonyi_at_ITS.TUDelft.NL
• FUZZ-IEEE'99
• The 8th International Conference on Fuzzy Systems

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Outline

• Problem formulation
• Proposed fuzzy modeling strategy
• Constrained parameter estimation of fuzzy models
• Formulation of a priori knowledge in inequality
  constraints
• Example identification of a liquid level process
• Conclusions

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Problem formulation
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The new fuzzy modeling strategy
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TS fuzzy model as an LPV model

• Linear Parameter Varying (LPV) representation

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The constrained parameter estimation method

• Prior knowledge ? local and global linear
  equality and inequality constraints.
• Rule-consequents define a convex region
  (polytope).
• This polytope can be constrained by global linear
  constraints.
• The individual rule-consequents can be
  constrained separately local linear constraints.

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Graphical representation of the method
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Quadratic programming

• H and d contain the measured input-output data
• ? and ? represents the a priori knowledge based
  constraints

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Types of prior knowledge

• Sampling (global)
• Stability (global)
• Stationary gain (global or local)
• Open-loop settling time (global or local)

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Example Stationary gain

• Upper and lower bounds

• Inequality constraints for QP

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Example Liquid level process

• Process input Flow rate (0-100)
• Process outputLiquid level in the bottom tank
  (0-100)

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Identification data

• Process input

• Process output

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Prior knowledge

• Open-loop stability
• Kmin 0, Kmax 2.5
• Settling-time of the local models

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Identified models

• Model 1 No a priori knowledge was used
  (VAF99.74)
• Model 2 Prior knowledge on process stability and
  steady-state gain (VAF99.74)
• Model 3 Model 2 Settling time (VAF99.78)

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All 3 models - good dynamic performance
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but completely different local behavior
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Conclusions

• Transform a priori knowledge into constraints on
  model parameters.
• Limited data prior knowledge ?good model
• Useful for control based on LPV model
• Detailed prior knowledge is needed
• Future research MIMO systems