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LPV model identification:
overview and perspectivesMarco LoveraDipartimento di Elettronica e Informazione, Politecnico di Milano
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Outline
Introduction and motivation
LPV model classes:
state-space form
input-output form
Issues in input-output to state-space conversion
Overview of LPV model identification
Perspectives and conclusions
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Introduction
Modern methods for robust and gain scheduled controllerdesign call for advanced modelling and identificationtechniques
Critical issue: deriving models in which the dependencefrom operating point information and/or uncertain
parameters is explicit
Linear Parametrically Varying (LPV) models: a useful
modelling approach to bridge the gap betweenidentification and controller design
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LPV models
Linear Parameter Varying systems are a class of linear time-
varying systems.
In state space form they are described as
Time varying systems, the dynamics of which are functions
of a measurable, time varying parameter vector .
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LPV models
Common assumptions on parameter vector :
Component-wise bounded
Component-wise rate-bounded
The equations
describe a whole family of time-varying systems.
A specific time-varying system is defined once a realisation (t) is
chosen.
A given LPV system can give rise to very different behaviours!
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LPV models
Motivation for LPV models:
Models for LTI systems subject to time-varying uncertainty
robust control problems
Models for LTV systems or linearizations of non linear
systems along the trajectory of
gain scheduling control problems
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Discrete-time LPV models in state-space form
We will now focus on models of the type
and define various model classes depending on how
enters the system matrices.
Finally we will compare the various structures and try to
find similarities and transformations between them.
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Structure of LPV models state-space form
Affine parameter dependence (LPV-A):
Input-affine parameter dependence (LPV-IA):
only B and D are function of
A and C are constant
Rational parameter dependence (LPV-R):
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Structure of LPV models state-space form
Linear fractional transformation parameter dependence(LPV-LFT):
P
()
u(t)
w(t)
y(t)
z(t)
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Structure of LPV models state-space formLPV-IA and LPV-LFT
LPV-IA and LPV-LFT models are related to each other.
The compound state space matrix M can be written as:
where each of the Mt
's has the structure
and is therefore a low rank (rt
) matrix.
The value of rt can be recovered exactly using an SVD
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Structure of LPV models state-space formLPV-IA and LPV-LFT
Expressing each of the Mt's by means of a rank rtdecomposition as Mt=UtVt one can write M() as
The obtained form for the system matrices coincides with the
one obtained in the special case of an LFTwith D00=0:
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Structure of LPV models state-space formLPV-A and LPV-LFT
This does not hold anymore in the case of an LPV-A model,
for which the Mt's matrices turn out to be "full":
so the LPV-A LPV-LFT conversion is likely to require
approximations if one aims at a small size .
Similarly, if the Mt's have been identified, the low rank
condition will almost certainly not hold.
It will be necessary to truncate the computed factorisation.
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Discrete-time LPV models in input-output form
We will consider input-output models of the type
which are parameter-dependent extensions of discrete-time
LTI input-output models.
As for state-space models, the ais and bjs can be
Affine
Rational Linear Fractional
functions of the parameter vector .
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Issues in input-output to state-space conversion
In the LTI case it is well known that if the state space model
corresponds to the input-output model
then all the state space models defined by
where T is square and non singular, are equivalentto theoriginal one, in the sense that they give rise to the same
input-output behaviour.
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Issues in input-output to state-space conversion
In LTI system identification this motivates a number of
methods which
1. Perform an initial (possibly nonparametric) estimation ofthe input-output behaviour
2. Refine the initial estimate
1. either directly in state-space form
2. or in input-output form, followed by conversion to
state-space form
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Issues in input-output to state-space conversion
In the LPV case, the above notion of equivalence class does
not hold anymore, as LPV models are time-varying.
In discrete-time:
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Issues in input-output to state-space conversion
In the LPV case, the above notion of equivalence class does
not hold anymore, as LPV models are time-varying.
In continuous-time:
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What happens in the LPV case?
The state transformation matrix T will be parameterdependent
Therefore, classical LTI input-output and state-spaceequivalence notions should be used very carefully!
In particular:
The change of state-space basis will depend Iocallyonthe value of the parameters and on their rate of change
The choice of input-output vs state-space modelsshould be based on the eventual goal of theidentification exercise, as conversion is not trivial.
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A simple example(Toth et al., 2007)
Consider the switching system described by
Which can be equivalently written in LPV form as
We can derive input-output models for the system using both
representations.
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A simple example(Toth et al., 2007)
Deriving input-output models for the individual modes and
interpolating we get
while from the state-space form we arrive at
In this case only a delay is introduced.
The issue becomes more critical for higher order systems.
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LPV model identification: an overview
Two broad classes of methods can be defined:
Global approaches
a single experiment the parameter is also excited
a parameter-dependent model is directly obtained
Local approaches
multiple experiments constant parameter values
many LTI models are obtained, which have to be
interpolated
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Overview of the literature: global approach
Input/output models
(Bamieh and Giarre,1999 & 2002)
(Previdi and Lovera, 2003 & 2004)
(Toth et al., 2007 & 2008) State space models
(Nemani et al, 1995)
(Lee and Poolla, 1997 & 1999) (Lovera et al., 1998)
(Sznaier and Mazzaro, 2001 & 2003)
(Verdult and Verhaegen, 2002)
(Felici et al., 2007)
(van Wingerden and Verhaegen, 2008)
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(Bamieh and Giarre 1999 & 2002)
Model class
Parameter estimation: linear least squares
Main contribution: characterisation of persistency ofexcitation conditions for input-output LPV models
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(Bamieh and Giarre 1999 & 2002)
The model can be written in linear regression form by
defining the parameter matrix and the extended regressor
So that
Parameters can be estimated recursively using LMS or RLS.
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(Bamieh and Giarre 1999 & 2002)
Main result: persistence of excitation.
Assuming that the input is sufficiently rich to insure that tis PE in the above sense, what is needed is that thetrajectory of t visit N distinct points infinitely many times.
The rate at which t revisits each of these limit pointsshould not slow down.
Thus, these revisits are sufficient to ergodically extract thecorrelation data of t .
(P idi d L 2003 & 2004)
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Previdi and Lovera 2003:
NLPV model class, analogy with local model networks.
Previdi and Lovera 2004:
NLPV model class, separable least squares estimationalgorithm.
(Previdi and Lovera 2003 & 2004)From LPV to NLPV models
P
NL
u(t)
z(t)
y(t)
w(t)
(t)
NLPV model class:
feedback is dynamicand nonlinear in
Th NLPV d l f il
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The NLPV model family
We start from the SISO time-varying ARX model
where the coefficients are given by
Variable zplays the role of a scheduling variable:
note that z is NOT measured, but estimated form data!
Th NLPV d l f il (2)
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The scheduling variablez is defined as
where is a suitable parameter vector and is a
regression vector given by
Note that is also a function of the input scheduling
variable (if available).
The NLPV model family (2)
The NLPV identification problem
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The NLPV identification problem
The overall model can be written as
where
Identification problem: find and minimising
Separable Cost Function
(Toth et al 2007 & 2008)
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(Toth et al., 2007 & 2008)
Besides the above mentioned issues with input-output to
state-space conversion, it is argued that:
An LPV system can be viewed as a collection of localbehaviours (associated with constant parameter values)
The overall behaviour of the system is given by aninterpolation of the local behaviours
The interpolation function is in general a function of time-
shifted versions of the parameters
A method based on OBFs is proposed, in the case ofstaticinterpolating functions (Wiener-LPV).
31Overview of the literature: global approach
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Overview of the literature: global approach
Input/output models
(Bamieh and Giarre,1999 & 2002)
(Previdi and Lovera, 2003 & 2004)
(Toth et al., 2007 & 2008)
State space models
(Nemani et al, 1995)
(Lee and Poolla, 1997 & 1999)
(Lovera et al., 1998)
(Sznaier and Mazzaro, 2001 & 2003)
(Verdult and Verhaegen, 2002)
(Felici et al., 2007)
32(Nemani et al 1995)
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(Nemani et al., 1995)
Identification of single input LPV-LFT models with a scalarparameter
Assume the state vector of the system is available formeasurement
Both cases of noise free and noisy state measurement aretaken into account, together with process noise in thestate equation.
The problem is solved using RLS; the use of IV-RLS isalso proposed to deal with non-white noise in the statemeasurements.
33(Lovera et al 1998)
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(Lovera et al. 1998)
Identification of MIMO LPV-A models
No restrictive assumptions on the number of parameters
Possibly noisy state vector measurement available
Batch solution using IV least squares
34(Lee and Poolla 1997 & 1999)
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(Lee and Poolla, 1997 & 1999)
A maximum likelihood (ML) algorithm for the identificationof MIMO LPV-LFT models is proposed
The algorithm is based on PEM and is strongly related toclassical methods for the ML identification of ARMA andARMAX models
The computation of the gradient and of the hessian isperformed by means of (LPV) filtering operations
Major issue related to this algorithm: initialisation
35(Sznaier and Mazzaro, 2001 & 2003)
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(Sznaier and Mazzaro, 2001 & 2003)
Consider a model class of the form
with
where
the Np Gi(z) transfer functions are known,
Gnp(z) is a stable, norm-bounded operator
is a bounded measurement noise.
An approach is provided which allows to test consistency (in
the form of LMIs) of the a priorimodelling information with
the results (measured y and parameters) of experiments.
36Subspace-based methods
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Subspace based methods
Extensions to LPV systems of classical subspace modelidentification algorithms for LTI models
Model class
37Subspace LPV identification
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p
Approach: use subspace techniques to estimate the state
sequence from the available I/O data.
To this purpose we have to:
Define the data equation for the state and the output of
the LPV model
Prove that we can estimate (at least approximately) the
state sequence
38Implementation issues: dimensionality
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p y
The number of rows in the data matrices grows exponentially with theorder of the system;
This would limit the applicability of these techniques to SISO systemsof small order;
How can this be circumvented?
Solutions available in the literature:
use the RQ factorisation to select the dominant rows in thedata matrices and discard the rest
use kernel methods to compress the row spaces of the datamatrices.
39LPV model identification: an overview
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Two broad classes of methods can be defined:
Global approaches
a single experiment the parameter is also excited
a parameter-dependent model is directly obtained
Local approaches
multiple experiments constant parameter values
many LTI models are obtained, which have to be
interpolated
40Overview of the literature: local approach
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Local approach
(Steinbuch et al2003)
(Paymans et al2006)
(Toth et al2007)
(Lovera Mercere 2007)
41Issues with local approaches
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Numerical accuracy:
Local problems are often formulated using poorlyconditioned canonical forms
This may lead to ill-conditioning in the interpolation
Consistency of the interpolation procedure:
Care must be taken when interpolating LTI identifiedmodels
Input/output form
interpolating transfer function coefficients State space form
consistency of state space basis
42The method of Steinbuch et al.
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The algorithm (applicable to SISO or MISO models) can be
summarised as follows:
1. Local experiments are performed and nonparametricestimates of the local frequency response arecomputed
2. Transfer functions
are fitted to the local frequency responses usingnonlinear LS
43The method of Steinbuch et al.
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3. Each transfer function is converted to CanonicalControllability Form:
4. The free parameters of the local models are interpolated
5. The model is converted to LFT form
44The method of Steinbuch et al.: discussion
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Numerical issues: the CCF is ill conditioned, so theinterpolation step will be numerically very sensitive.
Method restricted to
low order models
without sensitive poles/zeros (lightly damped complex
conjugate poles)
State space interpolation: no guarantee that all local
models are in the same state-space basis.
45The method of Paijmans et al
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Local models are parameterised using poles, zeros andgain
and factored into first and/or second order subsystems
46The method of Paijmans et al
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Each local model is decomposed using the following rules:
A second order system is created for each pair ofc.c.poles
All pairs of c.c. zeros are added to existing secondorder systems
For each remaining real pole a first order system is
created The remaining real zeros are added to the first and
second order subsystems
Parameter-dependent poles and zeros loci are optimisedin order to fit the pole/zero maps of the local models.
47The method of Paijmans et al: discussion
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Limited to SISO systems
Interpolation step very critical requires manual
intervention
Constraints introduce to preserve affine parameter
dependence: B and D matrices of local models must beconstant.
48A balanced subspace identification approach
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Consider the MIMO linear parametrically-varying system
Assume that the results of P identification experiments are available,associated with the operation of the system near different values ofthe parameter vector p
The problem consists in determining a set of parameter dependentmatrices
which provide a good approximation of the system over theconsidered range of operating points
49Outline of the balanced subspace method
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The proposed method addresses numerical issues, as follows.
1. Linear discrete-time state space models are estimated for eachoperating point, using a frequency-domain SMI algorithm, e.g.,
(McKelvey 1995)
2. The identified models are balanced using the numerical algorithm of(Laub et al1987)
3. If necessary, the balanced models are converted to continuous-timeusing a bilinear transformation
4. The p-dependent model is obtained by interpolation of the state-space matrices of the local models, made possible by the propertiesof balanced realisations
5. The model can be eventually converted to LFT form.
50Balanced realisations: definitions
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Definition: the matrices
are the observability and controllability Grammians for the
discrete-time system
The state space realization is internally balanced if
where are the singular values of
51Balanced realisations: definitions
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Definition: a state space realisation is qrinternally balanced
if
The considered frequency-domain identification algorithm is
such that
So for a stable system and large M the identified model is qr
internally balanced.
52Balanced realisations: useful properties
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Some classical results on balanced realisations and thebalancing transformation TB (Moore 1981, Laub et al1987,
Kabamba 1985)
If the eigenvalues of the product of the reachability andobservability Gramians are distinct, the eigenvectors areuniquely determined up to sign, so TB is essentially unique
If the eigenvalues are distinct, if the system matrices aresmooth functions of the parameter p, then so is TB
direct interpolation of the obtained state-spacematrices is possible
53Sign correction and interpolation
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The balanced form is only unique up to a sign, so manualcorrection of sign switching might be necessary
Interpolation is a trivial exercise in LS estimation
Conversion to LFT form can be performed as previously
illustrated
The LFT may be further optimised in order to compensate
for SVD truncation.
54A numerical example
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-20
-15
-10
-5
0
5
10
15
20
25
30
Magnitude(dB)
100
101
102
10
-45
0
45
90
135
180
225
Phase(deg)
Bode Diagram
Frequency (rad/sec)
Consider the parameter-dependent system
where
p [0.1, 0.95]
55Numerical example: estimated matrix elements
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-400
-200
0
200
400
ElementsofA
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.5
0
0.5
1
1.5
2
ElementsofB
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2
-1
0
1
ElementsofC
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.4
0.5
0.6
0.7
D
p
56Numerical example: Hankel singular values
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
p
Hankelsingularvalues
57Numerical example: transfer function coefficients
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
-2
100
102
104
106
108
Numeratorcoefficients
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
0
102
104
106
108
Denomina
torcoefficients
p
58Perspectives - a personal view:on square pegs and round holes...
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A fact: LPV model identification hardly used at all in practice.
There might be many reasons for this...
The tools are not available in public domain
The methods require unrealistic assumptions
The obtained models do not match the
existing design methods and tools:
the square peg-round hole problem
of system identification!
59The big picture(courtesy DLR)
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The modelling process
as currently viewed in the field:
A smooth flow
from nonlinear simulation
to models ready for robust
analysis and synthesis
Whe
reisd
ataus
ed?
60The role of prior knowledge and data
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It is often suggested
to use data at the simulator
level: model calibration
Simulink
Dymola
MoCaVa (T. Bohlin)is this sensible?
Prior knowledge
Experimental data
61The big picture revisited: step 1
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Basic physical knowledge should be used to build an
underlying model Only uncertainty structure should be fixed at this stage
(Modelica annotations?).
Prior knowledge
62The big picture revisited: step 2
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The LFT extraction should emphasize:
Measurable time-varying parameters Uncertain parameters to be refined using data.
63The big picture revisited: step 3
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Parameter estimation should be performed at this stage:
To simplify the optimisation process To come closer to an analytically tractable problem.
Experimental data
64Whats missing to complete the picture?
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Automated LFT extraction: many people working on this
Solved a long time ago for a specific application (DLR)
Available for explicit models in Simulink (UC Berkeley)
Undergoing development in Open Modelica
Identifiability analysis for structured LPV-LFT models
Results available Not enough for complete and automated process
Validation process
65Concluding remarks
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An overview of the last 20 years in the field of LPV modelidentification has been provided
A discussion of the pros and cons of each approach hasbeen offered
Some personal views on what the future of the areashould be have been given.