Non-linear System Identification: Possibilities and Problems

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Non-linear System Identification Possibilities
and Problems

• Lennart Ljung
• Linköping University

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Outline

• The geometry of non-linear identification
  Projections and visualization
• Identification for control in a non-linear system
  world
• Ongoing work with Matt Cooper, Martin Enquist,
  Torkel Glad, Anders Helmersson, Jimmy Johansson,
  David Lindgren, and Jacob Roll

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Geometry of Nonlinear Identification

• An elementary introduction

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A Data Set
Output
Input
Input
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A Simple Linear Model
Red Model Black Measured
Try the simplest model y(t) a u(t-1) b
u(t-2) Fit by Least Squares m1arx(z,0 2
1) compare(z,m1)
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A Picture of the Model
Depict the model as y(t) as a function of u(t-1)
and u(t-2)
u(t-2)
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A Nonlinear Model
Try a nonlinear model y(t) f(u(t-1),u(t-2)) m2
arxnl(z,0 2 1,sigm) compare(z,m2)
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The Predictor Function
Identification is about finding a reliable
predictor function that predicts the next output
from previous measured
data

• General structure

Common/useful special case
of fixed dimension m (state, regressors)
Think of the simple case
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The Data and the Identification Process
The observed data ZNy(1),?(1),y(N),?(N) are N
points in Rm1
The predictor model is a surface in this space
Identification is to find the predictor surface
from the data
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Outline

• The geometry of non-linear identification
  Projections and visualization
• Identification for control in a non-linear system
  world

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Projections Examine the Data Cloud

• In the plot of the y(t),?(t) the model surface
  can be seen as a thin projection of the data
  cloud.
• Example Drained tank, inflow u(t), level y(t).
  Look at the points ? y(t), y(t), u(t) in 3D
• What we saw
• How to recognize a thin projection?

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Nonlinearities Confined to a Subspace

• Predictor model ytf(?t)vt , f Rm -gt R
  
• Multi-index structure f(?t)b?t g(S?t) g Rk
  -gt R
• S is a k-by-m matrix, klt m The non-linearity is
  confined to a k-dimensional subspace (SSTI)
• If k1, the plot yt-b?t vs S?t will show the
  nonlinearity g.
• How to find b, S and g?

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How to Find b, S and g?

• Predictor function f(?,?)b? g(S(?)?,?)
• ? contains b, ? and ?
• ? may parameterize g, e.g. as a polynomial
• ? may parameterize S e.g. by angles in Givens
  rotations
• This is a useful parametrization of f if the
  nonlinearity is confined to a lower-dimensional
  subspace
• Minimization of criterion

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Example Silver Box Data

• Silver box data . (NOLCOS Special session)
• Fit as above with 5 past y and 5 past u in ? and
  use k1 (22 parameters) (Sparse data!)
• yb? g(S(?)?,?) (S R10 -gt R)

Simulation fit 0.44 Fit for ANN (with 617
pars) 0.46
Confined nonlinearities could be a good way to
deal with sparsity
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More Serious Visualization

• The interaction between a user and computational
  tools is essential in system identification. More
  should be done with serious visualization of data
  and estimation results, projections etc.
• We cooperate with NVIS Norrköping Visualization
  and Interaction Studio, which has a state-of-the
  art visualization theater. For preliminary
  experiments we have hooked up the SITB with the
  visualization package AVS/Express

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Outline

• The geometry of non-linear identification
  Projections and visualization
• Identification for control in a non-linear system
  world

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Control Design
Nominal Model
Regulator
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Control Design
True System
Regulator
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Control Design
Model error model
Nominal Model
Regulator
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Control Design
Model error model
True system
Nominal Model
Regulator
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Control Design
Model error model
Nominal Model
Nominal closed loop system
Regulator
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Robustness Analysis

• All robustness analysis relies upon one way or
  another checking the model error model in
  feedback with the nominal closed loop system.
  Some variant of the small gain theorem

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Model Error Models

• ? y ymodel

u
?
uu
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Identification for Control

• Identifiction for control is the art and
  technique to design identification experiments
  and regulator design methods so that the model
  error model matches the nominal closed loop
  system in a suitable way

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Linear Case

• Linear model and linear system Means that the
  model error model is also linear.
• Much work has been done on this problem (Michel
  Gevers, Brian Anderson, Graham Goodwin, Paul van
  den Hof, ) and several useful results and
  insights are available.
• Bottom line Design experimens so that model is
  accurate in frequency ranges where the stability
  margin is essential.
• Now for the case with nonlinear system .

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Non-linear System Approximation

• Given an LTI Output-error model structure
  yG(q,?)ue, what will the resulting model be for
  a non-linear system?
• Assume that the inputs and outputs u and y are
  such that the spectra ?u and ?yu are well
  defined.
• Then the LTI second order equivalent (LTI-SOE) is
  

Note G0 depends on u

• The limit model will be

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Example

• Consider the static system z(t)? u3(t)
• Let u(t) v(t)-2cv(t-1)c2v(t-2) where v is
  white noise with uniform distribution
• Then the LTI equivalent of the system is
• Note (1) Dynamic! (2) Static gain
  (?0.01,c0.99) 233

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Additivity of LTI-SOE

• Note that the LTI equivalent is additive (under
  mild conditions)

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Simulation
Blue without NL term Red With NL term
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Bode Plot

• Blue Estimated (LTI equivalent) model
• Green Linear part

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Lesson from the Example

• So, the gain of the model error model for ult1
  is 0.01 if the green linear model is chosen.
• And the gain of the model error model is (at
  least) 230 if the blue linear model is chosen.
• Unfortunately, System Identification will yield
  the blue model as the nominal (LTI-SOE) model!
• Lesson 1 The LTI-SOE linear model may not be
  the nominal linear model you should go for!

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Gain of Model Error Models

• Idea 1
• Traditional definition, possible problems with
  relay effect in the origin
• Idea 2 Affine power gain

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Model Error Model Gain

• So go for
• For all u?
• Impossible to establish
• Very conservative, typically relative error 1 at
  best.
• Lesson 2 For NL MEM necessary to let
• Must consider non-linear regulator!

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Possible Result for Nonlinear System

• Nominal model, linear or nonlinear
• Design an H1 non-linear regulator with the
  constraint
• and gain ? from output disturbance to
  controlled variable z
• The model error model obeys
• Then
• Where V(x(0)) is the loss for the nominal
  closed loop system

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Conclusions

• Geometry of non-linear identification
  Projections and visualization
• Identification for control with non-linear
  systems
• LTI-SOE may not be the best model
• Non-linear control synthesis necessary even with
  linear nominal model

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Epilogue

• Four Challenges for the Control Community
• A working theory for stability of black-box
  models.
• Prediction/Simulation
• Fully integrated software for modeling and
  identification
• Object oriented modeling
• Differential algebraic equations
• Full support of disturbance models
• Robust parameter initialization techniques
• Algebraic/Numeric
• Dealing with LTI-equivalents for good control
  design

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Global Patterns Lower Dimensional Structures

• In the linear case, experience shows that the
  data cloud often is concentrated to lower
  dimensional subspaces. This is the basis for PCA
  and PLS.
• Corresponding structure in the nonlinear case
• f(?)g(P?) P m n matrix, mltltn
• How to find P? (the multi-index regression
  problem)
• Note that sigmodial neural networks use basis
  functions fk?(?k? -?k) where ?? is a scalar
  product (ridge expansion). This is a similar
  idea (m1), that partly explains the success of
  these structures,

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More Flexibility
A more flexible, nonlinear model y(t)
f(u(t-1),u(t-2)) m3 arxnl(z,0 2
1,sigm,numb,100) compare(z,m3) compare(zv,m3)
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The Fit Between Model and Data
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Some Geometric Issues

• Look at the Data Cloud and figure out what may be
  good surface candidates (model structures)
• The cloud may be sparse.

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How to Recognize a Thin Projection?

• Idea 1 Measure the area of a collection of
  points by the area of its covariance ellipsoid
• SVD, Principal components, TLS etc Linear models

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How to Recognize a Thin Projection?

• Idea 1 Measure the area of a collection of
  points by the area of its covariance ellipsoid
• SVD, Principal components, TLS etc Linear models
• Idea 2 Delaunay Triangulation (Zhang)
• OK, but non-smooth criterion
• Idea 3 .

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How to Deal with Sparsity

• Sparsity Think of Johan Schoukenss Silver box
  data 120000 data points and 10 regressors
• Need ways to interpolate and extrapolate in the
  data space.
• Use Physical Insight Allow for few parameters to
  parameterize the predictor surface, despite the
  high dimension.
• Leap of Faith Search for global patterns in
  observed data to allow for data-driven
  interpolation.