subspace state-space system identificationisis j. c. gómez 3 they don’t require a particular...

ISIS J. C. Gómez 1

Subspace State-Space SystemIDentification

SSubspaceubspace SStatetate--SSpacepace SSystemystemIDIDentificationentification

ISIS J. C. Gómez 2

Subspace State-Space System IDentificationSSubspaceubspace SStatetate--SSpacepace SSystemystem IDIDentificationentification

They combine tools of System TheorySystem Theory, Numerical Linear AlgebraNumerical Linear Algebraand GeometryGeometry (projections).

They have their origin in Realization TheoryRealization Theory as developed in the 60/70s (Ho & Kalman, 1966).

They provide reliable state-space models of multivariablemultivariable LTI systems directlydirectly from input-output data.

They don’t require iterative optimization procedures no problems with local minima, convergence and initialization.

4SID Methods4SID Methods

PropertiesProperties

ISIS J. C. Gómez 3

They don’t require a particular (canonical) state-space realization numerical conditioning improves.

They require a modest computational load in comparison to traditional identification methods like PEM.

The algorithms can be (they have been) efficiently implemented in software like MatlabMatlab.

Main computational tools are QR and SVD.

All subspace methods compute at some stage the subspacesubspacespanned by the columns of the extended observability matrix.

The various algorithms (e.g., N4SID, MOESP, CVA) differ in the way the extended observability matrix is estimated and also in the way it is used to compute the system matrices.

ISIS J. C. Gómez 4

The system modelThe system model

kkkk

kkkk

eDuCxyKeBuAxx

++=++=+1 StateState--space model in space model in

innovation forminnovation form

The identification problemThe identification problemTo estimate the system matrices (A, B, C, D) and K , and the model order n, from an (N+α-1)-point data set of input and output measurements

{ } 11, −+=αN

kkk yu

ISIS J. C. Gómez 5

RealizationRealization--based 4SID Methodsbased 4SID Methods

∑∞

=−=

0kk uhy

⎩⎨⎧

>=

= − 0,0,

1BCAD

h

For a LTI system, a minimalminimal state-space realization (A, B, C, D)completely defines the input-output properties of the system through

where the impulse response coefficients are related to the system matrices by

convolution sumconvolution sum

h

ISIS J. C. Gómez 6

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

−++

+

11

132

21

jiii

j

j

ij

hhh

hhhhhh

H

Impulse Response Impulse Response HankelHankel MatrixMatrix

jiijH CΓ=

An estimate of the extended observability matrix can be computed by a full rankfull rank factorization of the impulse response Hankel matrix. This factorization is provided by the SVD of matrix Hij.

Extended Extended ObservabilityObservability

MatrixMatrix((I >n)I >n)

Extended Extended ControlabilityControlability

MatrixMatrix(j >n)(j >n)

ISIS J. C. Gómez 7

[ ]

ji

TTT

T

ij VUVUVVUUH

C

12

1

1

ˆ

21

111112

1

2

121 0

0⎟⎠⎞⎜

⎝⎛Σ⎟⎠⎞⎜

⎝⎛ Σ=Σ≈⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡Σ

Σ=

Γ

In the absence of noise, Hij will be a rank n matrix, and Σ1 will contain the n non-zero singular values →→ model order is model order is compucompu--tedted. . In the presence of noise, Hij will have full rank and a rank reduction stage will be required for the model order determination.

rank reductionrank reduction

ISIS J. C. Gómez 8

Computation of the system matricesComputation of the system matrices

Given estimates and of the extended observabilitymatrix, and the extended controllability matrix, respectively, estimates of the system matrices can be computed as:

îΓ

• : first row block of

• : first column block of

• : solving in the least squares sense

C îΓ

A

î i AΓ = Γ shiftshift--invariance propertyinvariance property

• 0D h=

B

C j

C j

ISIS J. C. Gómez 9

Problems:Problems: it is necessary to measure or to estimate (for example, via correlation analysis) the impulse response of the system →→ not not goodgood

ISIS J. C. Gómez 10

Direct 4SID MethodsDirect 4SID Methods

ααααα NUXY ++Γ= H

1 2

2 3 1

1 1

N

N

N

y y yy y y

y y y

α

α α α

+

+ + −

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

Y

Output block Output block HankelHankel matrixmatrix

(In a similar way are defined the Input block Hankel matrix Uα and the Noise block Hankel matrix Nα.)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=Γ

−1α

α

CA

CAC

ExtendedExtended (α > n) ObservabilityObservability MatrixMatrix

[ ]Nxxx ,,, 21=X

State Sequence MatrixState Sequence Matrix

(1)fundamental equationfundamental equation


⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

−−− DBCABCABCA

DCBCABDCB

D

H

432

000000

ααα

α

Lower triangular block Lower triangular block ToeplitzToeplitz matrix of impulse matrix of impulse responsesresponses (unknown).

The main idea of Direct 4SID methodsThe main idea of Direct 4SID methodsIn the absence of noise (Nα = 0), eq. (1) becomes

and the part of the output which does not emanate from the state can be removed by multiplying (from the right) both sides of eq. (2) by the orthogonal projection onto the null space of Uα, i.e. by

αααα UXY H+Γ= (2)


( ) ⊥Δ−Δ

⊥ =−=Π αααααα

UUUUUU

1TTIT orthogonal projectionorthogonal projection

0=⊥ααUU

This yields

⊥⊥ Γ= αααα XUUY

Note Note that the matrix on the left depends exclusively on the input-output data. Then, a full rank factorizationfull rank factorization of this matrix will provide an estimate of the extended observability matrix. Estimates of the corresponding system matrices can be obtained by resorting to the shift invariance property shift invariance property of the extended observability matrix, and by solving a system of linear equations in the least squares sense.

(3)

αΓ

such that


The factorization is provided by the SVD of the matrix on the left side

[ ] ⎟⎠⎞⎜

⎝⎛Σ⎟⎠⎞⎜

⎝⎛ Σ=Σ≈⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡Σ

Σ=

Γ

⊥ TTT

T

VUVUVV

UU 12

1

1

ˆ

21

111112

1

2

121 0

0

α

ααUY

rank reductionrank reduction(model order estimation)(model order estimation)

(In the absence of noise Σ2 = 0)

(4)


Computation of the system matricesComputation of the system matrices

Given an estimate of the extended observability matrix, estimates of the system matrices can be computed as:

αΓ

• : first row block of

• : solving in the least squares sense

C αΓ

A

Aαα Γ=Γ shiftshift--invariance propertyinvariance property

• solving a system of linear equations :ˆ and ˆ DB

Pre-multiply (2) by and post-multiply it by 2TU

( ) 1† T TU U U Uα α α α

−=


Equation (2) becomes

1

† † †2 2 2

0

ˆT T T

U I

U Y U U XU U H U Uα α α α α α α

= =

=

= Γ +

Linear equations on B and DLinear equations on B and D†

2 2T TU Y U U Hα α α=


In the presence of noise

ααααα NUXY ++Γ= H

⊥⊥⊥ +Γ= αααααα UNXUUYand

noise term needs to noise term needs to be removedbe removed

The noise term can be removed by correlating it awaycorrelating it away with a suitable matrix (Instrumental Variable). This can be interpreted as an oblique projection.oblique projection.

Presence of noisePresence of noise


Let us partition the input and output Hankel matrices into the past andfuture parts (somewhat arbitrary names !!)

For this to work, the noise variables in the output must be uncorrelatedwith the Instrumental Variables (IVs).

1 2

2 3 1

1 1

1 2

1 1

P

F

N

N

N

N

N

y y yy y y

y y yy y y

y y y

β β βα

β β β

α α α

+

+ + −

+ + +

+ + −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥= = ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

YY

Y


The model for the future outputs becomes

F F F F F FH= Γ + +Y X U N

The first step is to eliminate the input by post-multiplying by

(orthogonal projection)F⊥U

F F F F F F F⊥ ⊥ ⊥= Γ +Y U X U N U

Noting that the noise is not correlated to the inputs, and the futureinputs are not correlated to the past outputs, good candidates forthe IVs are the past inputs and outputs.


Under the assumption of ergodicity, it can be proved that

1lim 0

1lim 0

TF F PN

TF F PN

N

N

⊥

→∞

⊥

→∞

=

=

N U U

N U Y

which imply that the IV matrix

P

P

⎡ ⎤= ⎢ ⎥⎣ ⎦

UP

Y

can be used to asymptotically decorrelate the noise from F F⊥Y U


0

1 1 1lim lim limT T TF F F F F F FN N NN N N

⊥ ⊥ ⊥

→∞ →∞ →∞

=

= Γ +Y U P X U P N U P

The signal subspace can then be estimated consistently from the n principal left singular vectors of matrix

1 TF FN

⊥Y U P

1 1lim limT TF F F F FN NN N

⊥ ⊥

→∞ →∞= ΓY U P X U P (5)


Weighting MatricesWeighting Matrices

Row and column weighting matrices can be introduced in (5) before performing the SVD of the matrix in the left hand side. Any choice of positive-definite weighting matrices Wr and Wc will result in consistent estimates of the extended observability matrix.

[ ] ( )( )1 11 1 2 21 2 1 1 1 1 1 1 1

2 2ˆ

010

F

TT T T

r F F c T

VW W U U U V U V

N V⊥

Γ

Σ ⎡ ⎤⎡ ⎤= ≈ Σ = Σ Σ⎢ ⎥⎢ ⎥Σ⎣ ⎦ ⎣ ⎦

Y U P

change of coordinates in statechange of coordinates in state--spacespace


Existing algorithms employ different choices for matrices Wr andWc ,

• MOESP (Verhaegen, 1994):

• CVA (Larimore, 1990):

• N4SID (Van Overschee and de Moor, 1994):

121, T

r c FW I WN

−⊥⎛ ⎞= = ⎜ ⎟

⎝ ⎠PU P

1 12 21 1, T T

r F F F c FW WN N

− −⊥ ⊥⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠Y U Y PU P

11 21 1, T Tr c FW I W

N N

−⊥⎛ ⎞ ⎛ ⎞= = ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠PU P PP

subspace state-space system identificationisis j. c. gómez 3 they don’t require a particular...

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