[2008] LPV Model Identification

8/3/2019 [2008] LPV Model Identification

1/65

LPV model identification:

overview and perspectivesMarco LoveraDipartimento di Elettronica e Informazione, Politecnico di Milano


2/65

2

Marco Lovera

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


3/65

3

Marco Lovera

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


4/65

4

Marco Lovera

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 .


5/65

5

Marco Lovera

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!


6/65

6

Marco Lovera

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


7/65

7

Marco Lovera

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.


8/65

8

Marco Lovera

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):


9/65

9

Marco Lovera

Structure of LPV models state-space form

Linear fractional transformation parameter dependence(LPV-LFT):

P

()

u(t)

w(t)

y(t)

z(t)


10/65

10

Marco Lovera

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


11/65

11

Marco Lovera

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:


12/65

12

Marco Lovera

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.


13/65

13

Marco Lovera

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 .


14/65

14

Marco Lovera


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.


15/65

15

Marco Lovera


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


16/65

16

Marco Lovera


In the LPV case, the above notion of equivalence class does

not hold anymore, as LPV models are time-varying.

In discrete-time:


17/65

17

Marco Lovera


In the LPV case, the above notion of equivalence class does

not hold anymore, as LPV models are time-varying.

In continuous-time:


18/65

18

Marco Lovera

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.


19/65

19

Marco Lovera

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.


20/65

20

Marco Lovera

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.


21/65

21

Marco Lovera

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


22/65

22

Marco Lovera

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)


23/65

23

Marco Lovera

(Bamieh and Giarre 1999 & 2002)

Model class

Parameter estimation: linear least squares

Main contribution: characterisation of persistency ofexcitation conditions for input-output LPV models


24/65

24

Marco Lovera


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.


25/65

25

Marco Lovera


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)


26/65

26

Marco Lovera

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


27/65

27

Marco Lovera

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)


28/65

28

Marco Lovera

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


29/65

29

Marco Lovera

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)


30/65

30

Marco Lovera

(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


31/65

31

Marco Lovera

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)


(Verdult and Verhaegen, 2002)

(Felici et al., 2007)

32(Nemani et al 1995)


32/65

32

Marco Lovera

(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)


33/65

33

Marco Lovera

(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)


34/65

34

Marco Lovera

(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)


35/65

35

Marco Lovera


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


36/65

36

Marco Lovera

Subspace based methods

Extensions to LPV systems of classical subspace modelidentification algorithms for LTI models

Model class

37Subspace LPV identification


37/65

37

Marco Lovera

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


38/65

38

Marco Lovera

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


39/65

Marco Lovera

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


40/65

Marco Lovera

Local approach

(Steinbuch et al2003)

(Paymans et al2006)

(Toth et al2007)

(Lovera Mercere 2007)

41Issues with local approaches


41/65

Marco Lovera

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.


42/65

Marco Lovera

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.


43/65

Marco Lovera

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


44/65

Marco Lovera

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


45/65

Marco Lovera

Local models are parameterised using poles, zeros andgain

and factored into first and/or second order subsystems

46The method of Paijmans et al


46/65

Marco Lovera

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


47/65

Marco Lovera

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


48/65

Marco Lovera

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


49/65

Marco Lovera

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


50/65

Marco Lovera

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


51/65

Marco Lovera

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


52/65

Marco Lovera

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


53/65

Marco Lovera

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


54/65

Marco Lovera

-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


55/65

Marco Lovera

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


56/65

Marco Lovera

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


57/65

Marco Lovera

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...


58/65

Marco Lovera

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)


59/65

Marco Lovera

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


60/65

Marco Lovera

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


61/65

Marco Lovera

Basic physical knowledge should be used to build an

underlying model Only uncertainty structure should be fixed at this stage

(Modelica annotations?).

Prior knowledge



62/65

Marco Lovera

The LFT extraction should emphasize:

Measurable time-varying parameters Uncertain parameters to be refined using data.



63/65

Marco Lovera

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?


64/65

Marco Lovera

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


65/65

Marco Lovera

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.

Documents

[2008] LPV Model Identification