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Lennart LjungNon-linear System Identification
Leuven Workshop, KULOctober 12, 2004
Non-linear System Identification: Possibilities and Problems
Lennart Ljung
Linköping University
Lennart LjungNon-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
Ongoing work with Matt Cooper, Martin Enquist, Torkel Glad, Anders Helmersson, Jimmy Johansson, David Lindgren, and Jacob Roll
Lennart LjungNon-linear System Identification
Leuven Workshop October 12, 2004
Geometry of Nonlinear Identification
An elementary introduction
Lennart LjungNon-linear System Identification
Leuven Workshop October 12, 2004
A Data Set
InputOutput
Input
Lennart LjungNon-linear System Identification
Leuven Workshop October 12, 2004
A Simple Linear Model
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)
Red: Model Black: Measured
Lennart LjungNon-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 LjungNon-linear System Identification
Leuven Workshop October 12, 2004
A Nonlinear ModelTry a nonlinear model
y(t) = f(u(t-1),u(t-2))
m2 = arxnl(z,[0 2 1],’sigm’)
compare(z,m2)
Lennart LjungNon-linear System Identification
Leuven Workshop October 12, 2004
The Predictor Function
General structure
Common/useful special case:
Think of the simple case
of fixed dimension m (”state”, ”regressors”)
Identification is about finding a reliable predictorfunction that predicts the next output
from previous measured data
Lennart LjungNon-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
Identification is to find
the predictor surface
from the data:
The predictor model
is a surface in this space
Lennart LjungNon-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 LjungNon-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 LjungNon-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)=bt + g(St) 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-bt vs St will show the nonlinearity g.
How to find b, S and g?
Lennart LjungNon-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 LjungNon-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 nonlinearitiescould be a good way to deal with sparsity
Lennart LjungNon-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 LjungNon-linear System Identification
Leuven Workshop October 12, 2004
Lennart LjungNon-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 LjungNon-linear System Identification
Leuven Workshop October 12, 2004
Control Design
Nominal Model
Regulator
Lennart LjungNon-linear System Identification
Leuven Workshop October 12, 2004
Control Design
True System
Regulator
Lennart LjungNon-linear System Identification
Leuven Workshop October 12, 2004
Control Design
Nominal Model
Regulator
Model error model
Lennart LjungNon-linear System Identification
Leuven Workshop October 12, 2004
Control Design
Nominal Model
Regulator
Model error modelTrue
system
Lennart LjungNon-linear System Identification
Leuven Workshop October 12, 2004
Control Design
Nominal Model
Regulator
Model error model
Nominal closed loop
system
Lennart LjungNon-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 LjungNon-linear System Identification
Leuven Workshop October 12, 2004
Model Error Models
= y – ymodel
uu
u
Lennart LjungNon-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 LjungNon-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 LjungNon-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
The limit model will be
Note: G0 depends on u
Lennart LjungNon-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 LjungNon-linear System Identification
Leuven Workshop October 12, 2004
Additivity of LTI-SOE
Note that the LTI equivalent is additive (under mild conditions):
Lennart LjungNon-linear System Identification
Leuven Workshop October 12, 2004
Simulation
Output
0.01*u(1)^3
Nonlinear term
2z-1
z -z+.22
DiscreteTransfer Fcn
Band-LimitedWhite Noise
Blue: without NL term
Red: With NL
term
Lennart LjungNon-linear System Identification
Leuven Workshop October 12, 2004
Bode Plot
Blue: Estimated (LTI equivalent) model Green: ”Linear part”
Lennart LjungNon-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 LjungNon-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 LjungNon-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 LjungNon-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 LjungNon-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 LjungNon-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 LjungNon-linear System Identification
Leuven Workshop October 12, 2004
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,
Global Patterns: Lower Dimensional Structures
Lennart LjungNon-linear System Identification
Leuven Workshop October 12, 2004
More FlexibilityA more flexible, nonlinear modely(t) = f(u(t-1),u(t-2))m3 = arxnl(z,[0 2 1],’sigm’,’numb’,100)compare(z,m3)compare(zv,m3)
Lennart LjungNon-linear System Identification
Leuven Workshop October 12, 2004
The Fit Between Model and Data
Lennart LjungNon-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 LjungNon-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 LjungNon-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 LjungNon-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 Leap of Faith: Search for global patterns in observed data to allow for data-driven interpolation.to allow for data-driven interpolation.