Non-linear System Identification

Lennart LjungNon-linear System Identification

Leuven Workshop, KULOctober 12, 2004

Non-linear System Identification: Possibilities and Problems

Lennart LjungLinköping University


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



Geometry of Nonlinear Identification

An elementary introduction



A Data Set

InputOutput

Input



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



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)



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)



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



The Data and the Identification ProcessThe 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



Outline





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?



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?



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: …



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



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:





Outline





Control Design

Nominal Model

Regulator



Control Design

True System

Regulator



Control Design

Nominal Model

Regulator

Model error model



Control Design

Nominal Model

Regulator

Model error modelTrue

system



Control Design

Nominal Model

Regulator

Model error model

Nominal closed loop

system



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



Model Error Models

= y – ymodel

uu

u



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



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 ….



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



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



Additivity of LTI-SOE

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



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



Bode Plot

Blue: Estimated (LTI equivalent) model Green: ”Linear part”



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!



Gain of Model Error Models

Idea #1:

Traditional definition, possible problems with relay effect in the origin

Idea #2: Affine power gain



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!



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



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



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



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



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)



The Fit Between Model and Data



Some Geometric Issues

Look at the Data Cloud and figure out what may be good surface candidates (model structures)

The cloud may be sparse.



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



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: ….



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.

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Non-linear System Identification