Online Adaptive Parameter Estimator Design and Tuning

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Online Adaptive Parameter Estimator Design and
Tuning

• M.Shagalov and H.Budman
• Department of Chemical Engineering
• University of Waterloo
• Waterloo, Ontario, Canada

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Lecture Outline

• Neural networks in adaptive control and parameter
  estimation problems
• Gradient Descent technique
• Case study 1st order linear discrete system
• Stability based on Lyapunov theory
• Lyapunov function decay maximization with optimal
  adaptation gain tuning
• Simulation results summary and further work
  perspective

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Neural Networks in Parameter Estimation Problems

• Neural networks (NN) are widely used to estimate
  nonlinear dynamic models
• Applications adaptive estimation/control
• E.G radial basis function neural networks are
  linear with respect to adapting coefficients
  easy to proof stability and convergence
• Gradient descent method is used for coefficient
  adaptation

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First Order Linear Discrete System Adaptation
(Definitions)

• Plant
• Model
• Parameters (real,estimates,deviation)
• Output signal (measurement, prediction,
  deviation)

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Nonlinear RBF NN-based Model
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Adaptation Dynamics (Scalar Form)

• Error expression ek is defined as a function of
  parameter deviations present and past
  input/output measurements and adaptation gain KD
• Adaptation laws for parameter estimates dynamics

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Stability Based on Lyapunov Theory

• Quadratic Lyapunov function scalar form
• Stability is guaranteed when V is decreasing

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The Tuning Problem

• How to select the adaptation gains KD, Ka and Kb?
• Parameter estimates and error convergence for 2
  different KD values (Assuming Ka Kb1)

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Adaptation Dynamics (Matrix Form)

• Matrix Ak relates successive states to each other
  for both estimate and deviation forms

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Performance Based on Lyapunov Function Decrease

• Performance criteria is required for tuning
• Lyapunov function decrease parameter deviations
  and error convergence
• Since
• (function of actual unknown a priori
  parameters)
• it cannot be calculated explicitly

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Maximization of the Lyapunov Decay Rate

• State independent criteria for decay
• To maximize Lyapunov function decay over h time
  intervals
• Better performance may be achieved using the
  general criteria

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Analytical solution for a special case

• Solving for the eigenvalues of I- Ak T Ak
  analytical condition for KD tuning is obtained

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Overall LF Decrease
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Cases for Comparison

• Numerical optimal minimisation of the sum of
  Lyapunov over long horizon (only a posteriori
  known)
• Analytical optimal solution for past input/output
  data yk-1, uk-1
• Numerical optimal solution for past and current
  input/output data yk-1, uk-1, yk, uk
• Numerical optimal solution for past,current and
  future input/output data yk-1, uk-1, yk , uk ,
  yk1? ? yk1, uk1

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Simulation Results Summary

• Input/output data

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• Adaptation gain effect

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• Constant KD adaptation summary

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• Varying KD adaptation summary

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• Lyapunov function wrt time for the different
  cases

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• Lyapunov function wrt time for the different
  cases

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Concluding Remarks

• A methodology was developed for optimal KD tuning
  based on an eigenvalue norm
• Different cases based on the available
  input/output data measurements were compared
• For first order linear system with one step
  information an analytical solution is available
• The analytical solution is close to a poseriori
  calculated constant gain optimisation