Control Based on Instantaneous Linearization

1
Control Based on Instantaneous Linearization

• Eemeli Aro
• eemeli.aro_at_tkk.fi
• 16.11.2005

2
Structure

• Rationale
• Instantaneous linearization
• Controller implementation
• Discussion

3
Rationale

• Linear control better understood than nonlinear
  control
• Many well-established linear design techniques
  exist
• Linearization only valid in a limited operating
  range
• Extract a linear model from the current sample

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Instantaneous linearization

• Assume a deterministic neural network
  input-output model is available
• with a regression vector
• which is interpreted to define the state of the
  system

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Instantaneous linearization

• Then linearize g around ?(t?) to get an
  approximate model
• where

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Instantaneous linearization

• Separating components of the current regression
  vector ?(?) into a bias term ?(?)
• where

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Instantaneous linearization

• The coefficients ai and bi are collected into
  the polynomials
• Thus the approximate model can be seen as a
  linear model affected by a constant disturbance
  ?(?)

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Instantaneous linearization

• For a multilayer perceptron network with one
  hidden layer on tanh units and a linear output,

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Application to Control
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Application to Control

• At each sample, extract a linear model from a
  neural network model of the system and design a
  linear controller
• Can be seen as a gain scheduling controller with
  an infinite schedule

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Application to Control

• Structurally equivalent to an indirect
  self-tuning regulator, only difference is in how
  the linear model is extracted
• Control design based on certainty equivalence
  principle the controller is designed assuming
  that the linear model perfectly describes the
  system

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Application to Control

• Can implement any linear control design
• Need to compensate for bias term ?(?)
• e.g. by using integral action, which also
  compensates for other constant disturbances
• Need to keep in mind narrow operating range of
  linearized model

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Application to Control

• Pole placement design
• Assuming a linearized deterministic model
• The objective is to select the three polynomials
  R, S, and T so that the closed loop system will
  behave as

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Application to Control
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Application to Control

• Minimum variance design
• For regulation, not trajectory following
• Design the controller to minimize a criterion
  J(t)
• Generalized Minimum Variance controller
• where P, W and Q are rational transfer functions

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Discussion

• Pro
• Allows the use of linear design techniques
• Reasonably simple implementation
• Fast linearization design can be done between
  samples
• Allows control of systems with unstable inverses
  (with approximate pole placement controller
  design without zero cancellation)
• Can be used to understand the dynamics of the
  system (poles, zeros, damping, natural frequency)

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Discussion

• Con
• Linearized model often valid only in a narrow
  range
• Cant deal with hard nonlinearities
• Requires understanding on linear control theory

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References

• M. Nørgaard, O. Ravn, N. K. Poulsen, and L. K.
  Hansen, "Neural Networks for Modelling and
  Control of Dynamic Systems," Springer-Verlag,
  London, 2000
• O. Ravn, "The NNCTRL Toolbox. Neural networks for
  control", Version 2, Technical University of
  Denmark, 2003, http//www.iau.dtu.dk/research/cont
  rol/nnctrl.html