a brief motivation to study subspace system...

A brief motivation to studySubspace System Identification

Henri P. Gavin

Department of Civil & Environmental EngineeringDuke Univ.

2017-08-30

What is System Identification?

What is System Identification?

I the practice of extracting models from measurementsI a sub-set of parameter estimation specific to

dynamical systemsI modern methods (1990’s to today) (e.g., ERA and SSID)

• linear systems theory• geometric perpsective of projections

. . . as opposed to least sqaures• leverages bases of linear vector spaces computed from

matrix decompositions e.g., QR (LQ) and SVD• a projection of matrices of data onto space of models• characterized by direct matrix operations

. . . no iterative guess-and-check• multi-input, multi-output (MIMO) systems• the order of the system is contained in the data

. . . high order models (a dozen or more states)

Subspace System ID What is System Identification? CEE 629 H.P. Gavin 2017-08-30 1 / 44

Find a dynamic model from within the measurementsMeasurementssequence of inputs u(k)

sequence of outputs y(k)

Models

I MIMO FIR filters

y(k) =∑

i

H(i)u(k − i)

I state-space models

x(k + 1) = A x(k) + B u(k) + w(k)

y(k) = C x(k) + D u(k) + v(k)

I transfer functions

y(z) = H(z) u(z)

I nonlinearities . . . restricted to . . .u(k) = f(u(k))y(k) = g(y(k))

. . . x(k + 1) is linear in x(k)


dynamical systems


Geometric interpretation of ordinary least squares— fit a two-term polynomial to three data pointsI model equation ymodel = a1x + a2x2 ,I using values of the independent variable (error free, by assumption)

xT = [0.6,−0.7, 0.9] ... (and (x2)T = [0.36, 0.49, 0.81])I and using values of the dependent variable (measured, with some error)

yTdata = [0.2, 0.3, 1.2]

I model basis ... ymodel = [x, x2]a = Xa ,where the columns of X form a basis of the model.

I what are the OLS estimates for model parameters, a1, a2?I ordinary least squares parameter estimates ... a = [XT X ]−1(XT ydata)

I ordinary least squares model ...ymodel = Xa = (X [XT X ]−1XT )ydata = P⊥X ydata

where P⊥X is the orthogonal projection matrix onto the basis spanned by thecolumns of X . (It’s idempotent.)• The orthogonal projection of the data onto the space of models

minimizes the Euclidean length of ydata − ymodel .• The shortest distance from a point to a plane.


Geometric interpretation of ordinary least squares— fit a two-term polynomial to three data points

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1

y

x

data

modela1=0.223, a2=1.030

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.6-0.4-0.200.20.40

0.2

0.4

0.6

0.8

1

1.2

3

1

23

b1

b2

ydataymodel

err


projections and the identification of LTI system dynamics

I The outputs y are linear in the states x and the inputs u. y = Cx + DuI But the states are not measured!I This is the fundamental problem in LTI system ID.I How do we compute a state sequence corresponding to measured u and y?


oblique projection of data onto space of models

f

f

p

p

��

��

��

��

YU

UY

Xf

msmnterror

Y f

Yf

span U

Yp

p

span U f

D Uk f

P Xfk


Requirements

I persistent measured input . . .or an (assumed) persistent (unmeasured) noise disturbance

I the system responds persistently to persistent inputs

I the outputs do not feed back to inputs(there’s a way around this)


What could possibly go wrong?

. . . these problems are cousins . . .

I excessive measurement noise and/or process noise

I nearly linearly-dependent bases (ill-conditioning)

I over-parameterization

I non-linearity . . . Hammerstein systems


What to do about it.

. . . we will start the term examining some of these methods . . .

I pre-conditioning data (filtering, detrending) . . . but be careful!

I `2 regularization . . . bias and covariance

I `1 regularization (as a QP)

I rank reduction and total least squares

I structured rank reduction . . . a lot of opportunity here, I think.

I . . . nonlinearity . . . ??


What good is an identified LTI system. . . in an arbitrary basis?

I What is x? . . . It’s a good question!I If you don’t know what the identified state sequence x physically

corresponds to, can you do anything with the model? . . . e.g.,control?

I Maybe prediction is enough.I Maybe you know the mechanisms at play within the system

. . . physics-based modeling.I estimate (a few) physical parameters, e.g., L ,C ,R, K ,M,C,

from the identified LTI system dynamics, e.g., pj = −ζjωnj + iωd j

I usually a non-linear least squares problemLevenberg-Marquardt, etc.but with vastly reduced spaces of parameters and data!


System ID seems so essential! Why did ittake until now to figure out how to do itright?

Lineage

I Jacopo Riccati (1676 - 1754)I Alexandre-Theophile Vandermonde (1735 - 1796)I Carl Friedrich Gauss (1777 - 1855)I Camille Jordan (1838 - 1922)I Hermann Hankel (1839 - 1873)I Ferdinand Georg Frobenius (1849 - 1917)I Andrey Markov (1856 - 1922)I Aleksandr Lyapunov (1857 - 1918)I Issai Schur (1875 - 1941)I Otto Toeplitz (1881 - 1940)I Norbert Wiener (1894 - 1964)I Alston Scott Householder (1904 - 1993)I Rudolf E. Kalman (1930 - 2016)I Gene H. Golub (1932 - 2007)

Subspace System ID CEE 629 H.P. Gavin 2017-08-30 12 / 44

Subspace System Identification

Deterministic Subspace System Identificationdiscrete-time LTI in state-space

x(k + 1) = A x(k) + B u(k) + w(k) , x ∈ Rn , u ∈ Rr

y(k) = C x(k) + D u(k) + v(k) , y ∈ Rm

disturbance - noise covariance

E

( w(p)v(p)

)T (w(q)T v(q)T

) =

[Q SST R

]δpq

apply to sequences[

x(i + 1) · · · x(i + j)y(i) · · · y(i + j − 1)

]=

[A BC D

] [x(i) · · · x(i + j − 1)u(i) · · · u(i + j − 1)

]+

[w(i) · · ·w(i + j − 1)v(i) · · · v(i + j − 1)

][

Xi+1,jYi,j

]=

[A BC D

] [Xi,jUi,j

]+

[Wi,jVi,j

]

So, a realization A ,B ,C ,D may be obtained from input / output sequences with adirect matrix decomposition,if a state vector sequence [x(i) · · · x(i + j)] is known.

Subspace System ID Subspace System Identification CEE 629 H.P. Gavin 2017-08-30 13 / 44

Subspace ID : notation and data matricesSubstitute state equation into output equation, recursively, to obtain

y(i) · · · y(i + j − 1)y(i + 1) · · · y(i + j)...

.

.

.y(i + k − 1) · · · y(i + j + k − 2)

=

C

CA...

CAk−1

[

x(i) · · · x(i + j − 1)]

+

D 0 0 · · · 0CB D 0 · · · 0

CAB CB D · · · 0...

.

.

....

. . . 0CAk−2 CAk−3B CA k−4B · · · D

·

u(i) · · · u(i + j − 1)...

.

.

.u(i + k − 1) · · · u(i + j + k − 2)

+

Yi,j,k = Pk Xi,j +Dk Ui,j,k + Vi,j,k

Yp ≡ Y1,j,k Up ≡ U1,j,k Xp ≡ X1,j past I/O data and statesYf ≡ Yk+1,j,k Uf ≡ Uk+1,j,k Xf ≡ Xk+1,j future I/O data and states

Yp = Pk Xp +Dk Up + Vp

Yf = Pk Xf +Dk Uf + Vf


Subspace ID : three necessary assumptions

1. The input is persistent Ui,j,k has full row rank

2. The states respond persistently Xi,j has full row rank

3. There is no feedback u=⇒⇐=\ y span(Xi,j) ∩ span(Ui,j,k ) ∈ {Ø}

(there are ways around this)

So,

rank[

Up

Yp

]= kr + n

and any input / output sequence

[u(i) · · · u(i + j − 1)] , [y(i) · · · y(i + j − 1)]

is in the basis of [Up

Yp

]


Subspace ID : main lemma

span [Xf ] = span[

Up

Yp

]∩ span

[Uf

Yf

]

f

f

p

p

��

��

��

��

YU

UY

Xf

Xf lies in the span of[

Up

Yp

]Yf = Pk Xf +Dk Uf

Pk Xf lies in span[

Up

Yp

], and span

[Uf

Yf

],

Pk Xf is the oblique projectction of Yf into[

Up

Yp

]parallel to Uf .

msmnterror

Y f

Yf

span U

Yp

p

span U f

D Uk f

P Xfk


Subspace ID : computation

Compute the oblique projectcion via LQ decompositionUf[Up

Yp

]Yf

=

L11 0 0L21 L22 0L31 L32 L33

Q1

Q2

Q3

Pk Xf = L32 L−1

22

[Up

Yp

]Compute the state sequence via SVD

Pk Xf = UΣVT = UΣ1/2T · T−1Σ1/2VT

Pk = UnΣ1/2n T

Xf = T−1Σ1/2n VT

n =[

x(k + 1) · · · x(k + j)]

The state sequence is thus obtained, but in an arbitrary basis. . . . n4sid


primary reference texts for this course

I Katayama, Tohru, Subspace Methods for System Identification, Springer,2005.

I Van Overschee, Peter, and De Moore, Bart, Subspace Identification forLinear Systems, Kluwer, 1996.

I Juang, Jer-Nan, Applied System Identification, Prentice Hall, 1994.

Subspace System ID texts CEE 629 H.P. Gavin 2017-08-30 18 / 44

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&cad=rja&uact=8&ved=0ahUKEwjb5KTwiv_SAhVE7yYKHearAfsQFgg2MAQ&url=http%3A%2F%2Fiie.fing.edu.uy%2Fsimsee%2Fbiblioteca%2FSpringer%2C%2520Subspace%2520Methods%2520For%2520System%2520Identification%2520(2005)%2520Ddu%2520Lotb.pdf&usg=AFQjCNGBlLw-UwqHN8cNii1T7glbMUQO9g&sig2=y3jTLnnGNyX0jvKnSRQQig

http://www.shadmehrlab.org/Courses/learningtheory_files/subspace_book_96.pdf

http://www.shadmehrlab.org/Courses/learningtheory_files/subspace_book_96.pdf

http://www.pearsonhighered.com/educator/product/Applied-System-Identification/9780130792112.page

Some of the other topics in this course

Total Least Squares

I solve the over-determined noisy system

y ≈ Xa

I errors in y only . . . o.l.s.y + y = Xa

I errors in y and X . . . t.l.s.

y + y =[X + X

]a

I the t.l.s. problem

min∣∣∣∣∣∣∣∣[X y

]∣∣∣∣∣∣∣∣2F

such that y + y =[X + X

]a

I Eckart-Young thm.

Subspace System ID other topics CEE 629 H.P. Gavin 2017-08-30 19 / 44

OLS, `1, and `2 regularization

y(x; a1, a2) = a1x + a2x2 y = y + η ση = 0.5

OLS criterion mina ||y − Xa ||22`1 regularization criterion mina ||y − Xa ||22 + α||a ||1`2 regularization criterion mina ||y − Xa ||22 + β||a ||2

0

2

4

6

8

10

2 3 4 5 6 7 8 9

a2

a1

OLSL1L1L2L2

α = 20.0

α = 70.0

β = 1.0

β = 70.0

-5

0

5

10

15

0 0.2 0.4 0.6 0.8 1

true

msmnt

OLSL2L1


Nonlinear Least Squares - Levenberg Marquardt

0

50

100

150

200

-4 -2 0 2 4

|H(f

)|

f

Example 5(b)

0

50

100

150

200

-4 -2 0 2 4

|H(f

)|Example 5(a)


Singular Spectrum Analysis

I rank reduction of a Hankel matrix of a time series

Y =

y1 y2 · · · · · · yj−1 yj

y2 y3 · · · · · · yj yj+1...

......

...yk−1 yk · · · · · · yk+j−3 yk+j−2

yk yk+1 · · · · · · yk+j−2 yk+j−1

I truncated SVD expansion

Yn = UnΣnVTn

I reconstruct a Hankel form . . .average cross-diagonals (??) . . . or structured low-rank approximation

I extract a reduced basis from a very noisy measurement



10-1

100

100

101

102

σ i / σ1

i

n=10

s.v. ratio = 0.500

10-12

10-10

10-8

10-6

10-4

10-2

100

10-1

100

101

PSD

frequency, Hz

true

noisy

filtered

-1.5

-1

-0.5

0

0.5

1

1.5

10 12 14 16 18 20

signals

time, s


MIMO Wiener Filtering

I Estimate the (matrix-valued) Wiener coefficients (Markov parameters) H(i)

y(k) =∑

i

H(i) u(k − i)

I filter a noisy measurement from broad-band input-output dataI recover a reduced basis from a very noisy measurementI identify the impuse response of a MIMO system from noisy I/O data


Wiener filtering

-10

-5

0

5

10

0 10 20 30 40 50 60 70 80

mo

de

l p

red

ictio

n a

nd

tru

e s

ign

al

y+noisey

^y

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0.2 0.5 1 2 5 10 20 50p

ow

er

sp

ectr

al d

en

sity

frequency, Hz

N = 8000 SNR = 2.00 K = 20

Puu

Pyy

Pyy hat


MIMO Wiener Filter ID

-20

-15

-10

-5

0

5

10

15

20

0 1 2 3 4 5

outputs,

y

-40

-30

-20

-10

0

10

20

30

40

0 1 2 3 4 5

inputs, u

-20

-15

-10

-5

0

5

10

15

20

0 1 2 3 4 5

model p

rediction

(-) and t

rue outp

ut (o)

-15

-10

-5

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1 1.2

unit imp

ulse res

ponses

input # 2 -> output # 1

FFTtrue

identified


preliminaries needed for subspace identification

I linear vector spaces; decompositions; truncation; projections;ordinary least squares

I total least squares; SSA and Wiener filters

I nonlinear least squares; SQP; and `1 regularization

I linear time invariant systems, Kalman filter; controllability andobservability; Lyapunov equations; balanced realizations andmodel reduction;


Total Least Squares

I solve the over-determined noisy system

y ≈ Xa

I errors in y only . . . o.l.s.y + y = Xa

I errors in y and X . . . t.l.s.

y + y =[X + X

]a

I the t.l.s. problem

min∣∣∣∣∣∣∣∣[X y

]∣∣∣∣∣∣∣∣2F

such that y + y =[X + X

]a

I Eckart-Young thm.


OLS, `1, and `2 regularization

y(x; a1, a2) = a1x + a2x2 y = y + η ση = 0.5

OLS criterion mina ||y − Xa ||22`1 regularization criterion mina ||y − Xa ||22 + α||a ||1`2 regularization criterion mina ||y − Xa ||22 + β||a ||2

0

2

4

6

8

10

2 3 4 5 6 7 8 9

a2

a1

OLSL1L1L2L2

α = 20.0

α = 70.0

β = 1.0

β = 70.0

-5

0

5

10

15

0 0.2 0.4 0.6 0.8 1

true

msmnt

OLSL2L1


Nonlinear Least Squares - Levenberg Marquardt

0

50

100

150

200

-4 -2 0 2 4

|H(f

)|

f

Example 5(b)

0

50

100

150

200

-4 -2 0 2 4

|H(f

)|Example 5(a)



I rank reduction of a Hankel matrix of a time series

Y =

y1 y2 · · · · · · yj−1 yj

y2 y3 · · · · · · yj yj+1...

......

...yk−1 yk · · · · · · yk+j−3 yk+j−2

yk yk+1 · · · · · · yk+j−2 yk+j−1

I truncated SVD expansion

Yn = UnΣnVTn

I reconstruct a Hankel form . . .average cross-diagonals (??) . . . or structured low-rank approximation

I extract a reduced basis from a very noisy measurement



10-1

100

100

101

102

σ i / σ1

i

n=10

s.v. ratio = 0.500

10-12

10-10

10-8

10-6

10-4

10-2

100

10-1

100

101

PSD

frequency, Hz

true

noisy

filtered

-1.5

-1

-0.5

0

0.5

1

1.5

10 12 14 16 18 20

signals

time, s


MIMO Wiener Filtering

I Estimate the (matrix-valued) Wiener coefficients (Markov parameters) H(i)

y(k) =∑

i

H(i) u(k − i)

I filter a noisy measurement from broad-band input-output dataI recover a reduced basis from a very noisy measurementI identify the impuse response of a MIMO system from noisy I/O data


Wiener filtering

-10

-5

0

5

10

0 10 20 30 40 50 60 70 80

mo

de

l p

red

ictio

n a

nd

tru

e s

ign

al

y+noisey

^y

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0.2 0.5 1 2 5 10 20 50p

ow

er

sp

ectr

al d

en

sity

frequency, Hz

N = 8000 SNR = 2.00 K = 20

Puu

Pyy

Pyy hat


MIMO Wiener Filter ID

-20

-15

-10

-5

0

5

10

15

20

0 1 2 3 4 5

outputs,

y

-40

-30

-20

-10

0

10

20

30

40

0 1 2 3 4 5

inputs, u

-20

-15

-10

-5

0

5

10

15

20

0 1 2 3 4 5

model p

rediction

(-) and t

rue outp

ut (o)

-15

-10

-5

0

5

10

15

20

0 0.2 0.4 0.6 0.8 1 1.2

unit imp

ulse res

ponses

input # 2 -> output # 1

FFTtrue

identified


preliminaries needed for subspace identification

I linear vector spaces; decompositions; truncation; projections;ordinary least squares

I total least squares; SSA and Wiener filters

I nonlinear least squares; SQP; and `1 regularization

I linear time invariant systems, Kalman filter; controllability andobservability; Lyapunov equations; balanced realizations andmodel reduction;


Modeling Field Data

System Identification - Natural Periods

10-1

100

10-1

100

101

102

ide

ntifie

d n

atu

ral p

erio

ds,

s

peak foundation velocity, mm/sSubspace System ID Modeling Field Data CEE 629 H.P. Gavin 2017-08-30 37 / 44

System Identification - Damping Ratios

2

4

6

8

10

10-1

100

101

102

ide

ntifie

d d

am

pin

g r

atio

s,

\%

peak foundation velocity, mm/sSubspace System ID Modeling Field Data CEE 629 H.P. Gavin 2017-08-30 38 / 44

System Identification - Frequency Responses

101

102

100

101

sin

gu

lar

va

lue

sp

ectr

um

of

ide

ntifie

d t

ran

sfe

r fu

nctio

n

frequency (Hertz)Subspace System ID Modeling Field Data CEE 629 H.P. Gavin 2017-08-30 39 / 44

System Identification - modeling low-amplitude behavior

-15

-10

-5

0

5

10

30 32 34 36 38 40 42 44

top

, m

m/s

22011-11-18 23:11:14 UTC nMax=160, order=16, err=0.312

msmntsssid model

-15

-10

-5

0

5

10

30 32 34 36 38 40 42 44

ab

ove

iso

, m

m/s

2

msmntsssid model

Subspace System ID Modeling Field Data CEE 629 H.P. Gavin 2017-08-30 40 / 44

System Identification - modeling high-amplitude behavior

-1000

-500

0

500

1000

1500

20 25 30 35 40 45 50 55

top

, m

m/s

2

2011-12-23 02:17:47 - M 6.00 nMax=160, order=16, err=0.424

msmntsssid model

-1000

-500

0

500

1000

20 25 30 35 40 45 50 55

ab

ove

iso

, m

m/s

2

msmntsssid model


System Identification - Response PredictionRecord #114 Parameters Modeling Record #103 Output

15 20 25 30 35 40 45 50 55−2000

0

2000

acc, m

m/s

2

15 20 25 30 35 40 45 50 55−2000

0

2000

acc, m

m/s

2

15 20 25 30 35 40 45 50 55−2000

0

2000

acc, m

m/s

2

15 20 25 30 35 40 45 50 55−2000

0

2000

acc, m

m/s

2


System Identification - Response PredictionRecord #132 Parameters Modeling Record #114 Output

15 20 25 30 35 40 45 50 55 60−5000

0

5000

acc, m

m/s

2

15 20 25 30 35 40 45 50 55 60−5000

0

5000

acc, m

m/s

2

15 20 25 30 35 40 45 50 55 60−4000

−2000

0

2000

acc, m

m/s

2

15 20 25 30 35 40 45 50 55 60−2000

0

2000

4000

acc, m

m/s

2


System Identification and Response Predictionpre

dic

tion p

eak foundation v

elo

city, lo

g1

0(m

m/s

)

model peak foundation velocity, log10(mm/s)

1.2

1.4

1.6

1.8

2

2.2

1.2 1.4 1.6 1.8 2 2.2


a brief motivation to study subspace system...

Documents