fo for r th the co e coupled upled el elec ectric tric dri - - PowerPoint PPT Presentation

Ke Kern rnel el man anifold ifold re regress gression ion fo for r th the co e coupled upled el elec ectric tric dri rives ves dat ataset aset

Mirko ko Mazzolen leni Matteo Sca cande della lla Fabi bio Previd vidi

11 April 2018

mirko.mazzoleni@unibg.it matteo.scandella@unibg.it fabio.previdi@unibg.it

• 1. Introduction and motivation
• 2. A new framework for nonparametric system identification
• 3. Application to the coupled electric drive dataset
• 4. Conclusions and future developments

Ou Outline line

2/32

1.

• 1. Introd
• duc

uctio ion n and mo d motiva vation ion

• 2. A new framework for nonparametric system identification
• 3. Application to the coupled electric drive dataset
• 4. Conclusions and future developments

Ou Outline line

3/32

In Introduction

• duction an

and d mot

• tiv

ivation ation

4/32

Data Model Distance

In Introduction

• duction an

and d mot

• tiv

ivation ation

5/32

Data Mod

• del

el Distance

Mo Model del definition inition

In Introduction

• duction an

and d mot

• tiv

ivation ation

6/32

• Consider the NARX

RX system tem: 𝒯: 𝑧 𝑢 + 1 = 𝑕 𝑦𝑣 𝑢 , 𝑦𝑧(𝑢) + 𝑓 𝑢 , where:  𝑧 𝑢 ∈ ℝ is the system output  𝑕(𝑢) is a nonlinear inear function ction  𝑦𝑣 𝑢 = 𝑣 𝑢 , … , 𝑣 𝑢 − 𝑞 + 1

𝑈 ∈ ℝ𝑞×1 is a regressor vector of past 𝑞 inputs

 𝑦𝑧 𝑢 = 𝑧 𝑢 , … , 𝑧 𝑢 − 𝑟 + 1

𝑈 ∈ ℝ𝑟×1 is a regressor vector of past 𝑟 outputs

 𝑦 𝑢 = 𝑦𝑧 𝑢 , 𝑦𝑧 𝑢

𝑈 ∈ ℝ𝑞+𝑟×1

 𝑓 𝑢 ∈ ℝ is an additive white noise

Re Repr producing

• ducing Kernel

rnel Hilbert bert Spaces aces (R (RKHS) KHS)

Le Lear arning ning from

• m dat

ata

7/32

• An RKHS is a Hilber

bert space ce ℋ such that:

a. Its elements are functions 𝑣: Ω → ℝ , where Ω is a generic set b. ∀𝑦 ∈ Ω, 𝑀𝑦: ℋ → ℝ is a continuous linear functional

• Riesz’s repr

pres esentation entation theorem em  ∃ 𝑠

𝑦 ∈ ℋ s.t. 𝑀𝑦 𝑣 = 𝑣 𝑦 = 𝑣, 𝑠 𝑦

• The function 𝑠

𝑦(⋅) is called the repr

pres esenter enter of evaluatio aluation in 𝑦

𝑣 → 𝑣(𝑦)

Re Repr producing

• ducing Kernel

rnel Hilbert bert Spaces aces (R (RKHS) KHS)

Le Lear arning ning from

• m dat

ata

8/32

• The reproducing kernel is defined as: 𝐿 𝑦, 𝑨 =

𝑠

𝑦, 𝑠 𝑨 , 𝐿 = Ω × Ω → ℝ

a. Symmetric: 𝐿 𝑦, 𝑨 = 𝐿 𝑨, 𝑦 b. Semidefinite positive σ𝑗,𝑘=1

𝑜

𝑑𝑗𝑑

𝑘𝐿(𝑦𝑗, 𝑦𝑘) ≥ 0 ∀𝑜, 𝑑𝑗 ∈ ℝ, ∀𝑦𝑗 ∈ Ω

• Mo

Moor

• re-Ar

Aronsza

• nszajn

jn theo theorem  A RKHS KHS defines a corresponding repr eproducing

• ducing

kernel

• rnel. Conversely, a reproducing
• ducing kernel

rnel defines a unique RKH KHS

Exa xampl mples of kerne nels ls

Le Lear arning ning from

• m dat

ata

9/32

• Constant

tant kernel: rnel: 𝐿 𝑦, 𝑨 = 1

• Linear

ear kernel: rnel: 𝐿 𝑦, 𝑨 = 𝑦 ⋅ 𝑨

• Polyn

ynomial mial kernel el: : 𝐿 𝑦, 𝑨 = 𝑦 ⋅ 𝑨 + 1 𝑒

• Gauss

ssian an kernel: rnel: 𝐿 𝑦, 𝑨 = 𝑓− 𝑦−𝑨 2

2𝜏2

Kernel rnel comp mposi

• sition

tion th theorem eorem:

• A linear combination of valid kernel functions is a valid kernel function
• The space induced by this kernel is the span of the spaces induced by the single ones

𝐼 =⊕𝑗 𝐼𝑗

Kernel rnel me method thods in syste tem identi entific fication ation

A new ew fram amewo ework rk for

• r sy

syst stem em iden entifi tification cation

10 10/32 Stab able spline ine kernel el Represe resenters ters 𝑡

• Pillonetto, Gianluigi and Giuseppe De Nicolao. “A new kernel-based approach for linear system identification.” Automatica 46 (2010): 81-93.
• Pillonetto, Gianluigi et al. “A New Kernel-Based Approach for NonlinearSystem Identification.” IEEE Transactions on Automatic Control 56 (2011): 2825-2840.

In Introduction

• duction an

and d mot

• tiv

ivation ation

11 11/3 /32

Data Model Di Dist stan ance ce

No Nonpa parametric rametric learning rning

Le Lear arning ning from

• m dat

ata

12 12/32

• Consider the variational

riational formulation: ො 𝑕 = arg min

𝑕∈ℋ

෍

𝑗=1 𝑂

𝑧𝑗 − 𝑕 𝑦𝑗

2 + 𝜇𝑈 ⋅ 𝑕 ℋ 2

𝑧𝑗 = 𝑧 𝑢𝑗 ; 𝑦𝑗 = 𝑦 𝑢𝑗

• Tikhonov regularization: 𝑕 2

ℋ penalizes the norm of the fitted function

• The minimization problem is on the RKHS space ℋ  infinite number of parameters!

Re Representer presenter th theorem

• rem

Le Lear arning ning from

• m dat

ata

13 13/32

• The minimizer

imizer of the variational problem is given by:

• Linear comb

mbination ination of the representer esenters of the training points 𝑦𝑗, evaluated in the point 𝑦

• The solution is expressed as combination of «basis functions» which properties are

determined by ℋ ො 𝑕(𝑦) = ෍

𝑗=1 𝑂

𝑑𝑗𝐿 𝑦, 𝑦𝑗 = ෍

𝑗=1 𝑂

𝑑𝑗𝑠

𝑦𝑗(𝑦)

For some 𝑂-tuple 𝑑 = 𝑑1, 𝑑2, … , 𝑑𝑂 𝑈 ∈ ℝ𝑂×1

No Nonpa parametric rametric learning rning – Solution ution

Le Lear arning ning from

• m dat

ata

14 14/3 /32

• Using the representer theorem it possible to express the variational problem as:
• Since the expression is quadratic in 𝑑 we have the closed
• sed-fo

form solution ution:

Ƹ 𝑑 = arg min

𝑑∈ℝ𝑂

𝑍 − 𝒧𝑑 2

2 + 𝜇𝑈 ⋅ 𝑑𝑈𝒧𝑑

• 𝑍 ∈ ℝ𝑂×1: vector of observations
• 𝒧 ∈ ℝ𝑂𝑦𝑂: semidefinite positive and

symmetric matrix, s.t. 𝒧𝑗𝑘 = 𝐿(𝑦𝑗, 𝑦𝑘)

𝒧 + 𝜇𝑈 ⋅ 𝐽𝑂 ⋅ Ƹ 𝑑 = 𝑍

• 1. Introduction and motivation

2.

• 2. A new fr

frame mework work fo for nonpa parame rametric ric sy syst stem m ide dentif ific icati ation

• n
• 3. Application to the coupled electric drive dataset
• 4. Conclusions and future developments

Ou Outline line

15 15/32

Ma Manifol ifold d learning rning (s (sta tatic tic system tems) s)

In Introduction

• duction an

and d mot

• tiv

ivation ation

16 16/32

• Suppose that the regressors’ belong to a

manif ifold

• ld in the regressors’ space
• The position

ition of the regressors adds prio ior r information mation

• How to incorporate this information in a

learning ning framework? amework?

Ma Manifol ifold d learning rning (s (sta tatic tic system tems) s)

In Introduction

• duction an

and d mot

• tiv

ivation ation

17 17/32

• Suppose that the regressors’ belong to a

manif ifold

• ld in the regressors’ space
• The position

ition of the regressors adds prio ior r information mation

• How to incorporate this information in a

learning ning framework? amework?

Incorpor

• rporating

ating th the ma manifol fold d infor formation mation

In Introduction

• duction an

and d mot

• tiv

ivation ation

18 18/32

Semi Semi-su super pervi vise sed d smoothne thness ss assumption ption If two regressors 𝑦 𝑗 and 𝑦 𝑘 in a high-density region are close, then so should be their corresponding outputs 𝑧 𝑗 and 𝑧 𝑘

Li Link k to to dynamic namical al syste tems ms

In Introduction

• duction an

and d mot

• tiv

ivation ation

19 19/32

• In dynamical systems, regressors can

be stron

• ngly

gly corr rrelated elated

• It is meaningful to think that they lie

lie on a a manifold fold of the regressors’ space

• PCA reveals how 91% of variance

riance explained by one component

In Introduction

• duction an

and d mot

• tiv

ivation ation

20 20/32

Ma Manifol ifold regular ularizati ization

• n
• One way to enforce the smoothness

ness assumption ption is to minimize the quantity:

𝑇𝑕 = න

𝒣

𝛼 ⋅ 𝑕 2 𝑒𝑞 𝑦 = න

𝒣

𝑕 ⋅ Δ ⋅ 𝑕 𝑒𝑞 𝑦

• 𝑞 𝑦 : probability distribution of the regressors, 𝛼: Gradient, Δ: Laplacian-Beltrami operators
• Minimizing 𝑇𝑕 means mi

minim imizing izing th the gradient dient of the function

• The term can rarely be computed since 𝑞 𝑦 and 𝒣 are unkno

nown wn

In Introduction

• duction an

and d mot

• tiv

ivation ation

21 21/32

Ma Manifol ifold regular ularizati ization

• n
• The term 𝑇𝑕 can be modeled

led using the reg egres esso sor r graph ph, encoding connections and distasnces between points:  with the regressors as its vertices  the weights on the edges are defined as:

• Considering the Laplacian matrix 𝑀 = 𝐸 − 𝑋 ∈ ℝ𝑂×𝑂, where:

 𝐸 ∈ ℝ𝑂×𝑂 is a diagonal matrix 𝐸𝑗𝑗 = σ𝑘=1

𝑂

𝑥𝑗,𝑘; 𝑋 ∈ ℝ𝑂×𝑂 contains the 𝑥𝑗,𝑘 𝑥𝑗,𝑘 = 𝑓

− 𝑦 𝑗 −𝑦 𝑘

2

2𝜏𝑓

2

• 𝜏𝑓 is a tuning parameter
• Higher value  similar

regressors

A new ew fram amewo ework rk for

• r sy

syst stem em iden entifi tification cation

22 22/32

Ma Manifol ifold regular ularizati ization

• n
• We have therefore that:

𝑇𝑕 = න

𝒣

𝛼 ⋅ 𝑕 2 𝑒𝑞 𝑦 = න

𝒣

𝑕 ⋅ Δ ⋅ 𝑕 𝑒𝑞 𝑦 ≈ 𝐺𝑈 ⋅ 𝑀 ⋅ 𝐺

• The vector 𝐺 = 𝑕 𝑦 1

, ⋯ , 𝑕 𝑦 𝑂

𝑈 ∈ ℝ𝑂×1 contains the noisele

seless ss outputs

• T
• compute the approximation, only the regress

essor

• rs are needed

A new ew fram amewo ework rk for

• r sy

syst stem em iden entifi tification cation

23 23/32

Ma Manifol ifold regular ularizati ization

• n
• We can than enforce the smoothness assumption by adding a new regular

ulariza ization tion term rm:

• The representer theorem still

ll holds for this cost function ො 𝑕 = arg min

𝑕∈ℋ

σ𝑗=1

𝑂

𝑧(𝑢) − 𝑕 𝑦 𝑢

2

+ 𝜇𝑈 ⋅ 𝑕 ℋ

2 + 𝜇𝑁 ⋅ 𝐺𝑈 ⋅ 𝑀 ⋅F

• The solution to the minimization problem admits a closed

sed-fo form solutio tion:

𝒧 + 𝜇𝑈 ⋅ 𝐽𝑂 +𝜇𝑁 ⋅ 𝑀 ⋅ 𝒧 ⋅ Ƹ 𝑑 = Y 𝐵 ⋅ Ƹ 𝑑 = Y

• 1. Introduction and motivation
• 2. A new framework for nonparametric system identification

3.

• 3. App

pplica cation ion to the co coupl pled ed elect ctric ric dr drive e da datase set

• 4. Conclusions and future developments

Ou Outline line

24 24/32

In Introduction

• duction an

and d mot

• tiv

ivation ation

25 25/32

Data Model Distance

Application plication to co

• couple

pled d el elec ectric ric drive ive dat ataset aset

26 26/32

• Application to the coupled

pled electric ectric drives ives

• Input:

t: motor voltage Uniformely distributed dataset 𝑂 = 500

• Output:

t: pulley speed

• Ts = 20ms

Resul sults ts

Motor 1 Motor 2 Spring Flexible belt Pulley Transducer

Application plication to co

• couple

pled d el elec ectric ric drive ive dat ataset aset

27 27/32

Re Result sults

• The method has been applied to the coupled

pled electric ectric drives ives data taset set

• The following kernel

rnel was employed:

𝐿 𝑦, 𝑨 = 𝜃𝑜𝑚 ⋅ 𝑓− 𝑦−𝑨 2

𝜏2

+ 𝜃𝑚 ⋅ 𝑦𝑈𝑨 + 𝜃𝑑

Hype perpar parame ameter ters s 𝜔 = [𝜇𝑈, 𝜇𝑁, 𝜃𝑜𝑚, 𝜃𝑚, 𝜃𝑑 , 𝜏, 𝜏𝑓]

Application plication to co

• couple

pled d el elec ectric ric drive ive dat ataset aset

28 28/32

Re Result sults

• The hyperparamet

erparameter ers have been estim imated ated using Generalized Cross Validation (GCV):

• Where 𝜉 𝜔 are the degree

ee of freedo edom of the model:

• The order of the system is

is estimat imated ed via a grid search as: Ƹ 𝑞, ො 𝑟 = argmin

𝑞,𝑟 𝐾𝑞,𝑟 ෠

𝜔𝑞,𝑟 ෠ 𝜔𝑞,𝑟 = argmin

𝜔 𝑂 𝑂−𝜉 𝜔

2 ⋅ 𝑍 − ෠

𝑍 2

2 = argmin 𝜔 𝐾𝑞,𝑟 𝜔

𝜉 𝜔 = 𝑢𝑠 𝑇 ෠ 𝑍 = 𝒧 ⋅ Ƹ 𝑑 = 𝒧 ⋅ 𝐵−1 ⋅ 𝑍 = 𝑇 ⋅ 𝑍

Application plication to co

• couple

pled d el elec ectric ric drive ive dat ataset aset

29 29/32

• Si

Simulatio lation results ults improve over existing approaches

• All kernels equipped with linear

and constant terms

Resul sults ts

• Gianluigi Pillonetto, Minh Ha Quang, Alessandro Chiuso. “A New Kernel-Based Approach for NonlinearSystem Identification.” IEEE Transactions on Automatic

Control 56 (2011): 2825-2840

• 1. Introduction and motivation
• 2. A new framework for nonparametric system identification
• 3. Application to the coupled electric drive dataset

4.

• 4. Concl

clus usion ions s and fu d future re de develop lopme ment nts

Ou Outline line

30 30/32

Co Conclusi clusions

• ns an

and d future ure dev evel elopm

• pment

ents

31 31/32

• A new paradigm for nonparametric learning of nonlinear system was presented
• The idea leverages the concept of manifold regularization
• Future challenges (we are working on it already):

 Bayesia esian deriva ivatio tion of the manifold regularization approach  Application to wider range of systems and nonlinearities

Concl nclusi usions

• ns

Co Conclusi clusions

• ns an

and d future ure dev evel elopm

• pment

ents

32 32/32

• M. Mazzoleni, S. Formentin, M. Scandella, F. Previdi «Semi-supervised learning of dynamical systems: a

preliminary study.» 16th European Control Conference (ECC), Limassol, Cyprus, 2018. In press.

• M. Mazzoleni, M. Scandella, S. Formentin, F. Previdi «Identification of nonlinear dynamical system with

synthetic data: a preliminary investigation.» 18th IFAC Symposium on System Identification, Stockholm, Sweden, 2018. In press.

References ferences

Concl nclusi usions

• ns and futu

ture re develop velopments ments

TH THAN ANKS FO KS FOR R YO YOUR UR AT ATTEN TENTION TION