Non-linear System Identification:
Possibilities and Problems
Lennart Ljung
Linköping University
Lennart Ljung
Non-linear System Identification
Leuven Workshop, KUL
October 12, 2004
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
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Geometry of Nonlinear Identification
 An elementary introduction
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
A Data Set
Output
Input
Input
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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:
m1=arx(z,[0 2 1])
compare(z,m1)
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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)
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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)
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
The Data and the Identification Process
The observed data
ZN=[y(1),(1),…y(N),(N)]
are N points in Rm+1
The predictor model
is a surface in this
space
Identification is to find
the predictor surface
from the data:
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Outline
 The geometry of non-linear identification:
Projections and visualization
 Identification for control in a non-linear
system world
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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?
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Nonlinearities Confined to a Subspace
 Predictor model: yt=f(t)+vt , f: Rm -> R
 Multi-index structure: f(t)=bt + g(St) g: Rk -> R
 S is a k-by-m matrix, k< m: The non-linearity is confined to
a k-dimensional subspace (SST=I)
 If k=1, the plot yt-bt vs St will show the nonlinearity g.
 How to find b, S and g?
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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: …
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Example: Silver Box Data
 Silver box data: …. (NOLCOS Special session)
 Fit as above with 5 past y and 5 past u in  and use k=1:
(22 parameters) (Sparse data!)
y=b + g(S(),) (S: R10 -> R)
Simulation fit: 0.44
Fit for ANN
(with 617 pars): 0.46
Confined nonlinearities
could be a good way
to deal with sparsity
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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:
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Outline
 The geometry of non-linear identification:
Projections and visualization
 Identification for control in a non-linear
system world
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Control Design
Nominal Model
Regulator
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Control Design
True System
Regulator
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Control Design
Model error model
Nominal Model
Regulator
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Control Design
Model error model
True
system
Nominal Model
Regulator
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Control Design
Model error model
Nominal
closed
loop
system
Nominal Model
Regulator
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Model Error Models
 = y – ymodel
u

u
u
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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 ….
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Non-linear System Approximation
 Given an LTI Output-error model structure y=G(q,)u+e,
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
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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,c=0.99): 233
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Additivity of LTI-SOE
 Note that the LTI equivalent is additive (under mild
conditions):
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Simulation
2z-1
z2 -z+.2
Band-Limited
White Noise
Discrete
Transfer Fcn
Output
0.01*u(1)^3
Nonlinear term
Blue:
without
NL term
Red:
With NL
term
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Bode Plot
 Blue: Estimated (LTI equivalent) model
 Green: ”Linear part”
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Lesson from the Example
 So, the gain of the model error model for |u|<1 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!
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Gain of Model Error Models
 Idea #1:
 Traditional definition, possible problems with relay effect in the
origin
 Idea #2: Affine power gain
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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!
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Epilogue
 Four Challenges for the Control Community:
1) A working theory for stability of black-box models.
 Prediction/Simulation
2) Fully integrated software for modeling and identification
 Object oriented modeling
 Differential algebraic equations
 Full support of disturbance models
3) Robust parameter initialization techniques
 Algebraic/Numeric
4) Dealing with LTI-equivalents for good control design
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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, m<<n
 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 (m=1), that partly
explains the success of these structures,
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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)
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
The Fit Between Model and Data
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
Some Geometric Issues
 Look at the Data Cloud and figure out what may be good
surface candidates (model structures)
 The cloud may be sparse.
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
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: ….
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004
How to Deal with Sparsity
 Sparsity: Think of Johan Schoukens’s 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.
Lennart Ljung
Non-linear System Identification
Leuven Workshop
October 12, 2004