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Control Based on Instantaneous Linearization

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Title: 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

4
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

5
Instantaneous linearization
  • Then linearize g around ?(t?) to get an
    approximate model
  • where

6
Instantaneous linearization
  • Separating components of the current regression
    vector ?(?) into a bias term ?(?)
  • where

7
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
    ?(?)

8
Instantaneous linearization
  • For a multilayer perceptron network with one
    hidden layer on tanh units and a linear output,

9
Application to Control
10
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

11
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

12
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

13
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

14
Application to Control
15
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

16
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)

17
Discussion
  • Con
  • Linearized model often valid only in a narrow
    range
  • Cant deal with hard nonlinearities
  • Requires understanding on linear control theory

18
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
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