Subspace Identification of Structured State-Space Models...

Subspace Identification of Structured State-Space

Models with Unknown Inputs

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Chengpu Yu (|¤Ë)

Key Laboratory of Intelligent Control and Decision of Complex Systems

Beijing Institute of Technology, China

June 24, 2019

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Outline

1 Background

2 Problem formulation

3 Identifiability analysis

4 Subspace identification method

5 Numerical simulations

6 Conclusions

2 / 29

Outline

1 Background





6 Conclusions

3 / 29

Background

Structured state-space model: a spatio-temporal model with structured interconnectionin the space domain and dynamic response in the time domain.

Applications of the structured state-space model identification:

1 Provide network models for general implementation of networked systems.

2 Estimate parameters involved in physical models.

3 Approximate continuous physical objects by finite-element networks.

Figure: Multi-agent network Figure: Mass-spring model

Figure: Finite-element network

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Background

Traditional SysID methods require (noisy) input-output measurements; however, withunknown inputs, the identification problem (identifiability analysis + identificationmethod development) becomes challenging.

Examples of structured systems with unknown inputs are: local network system withunknown interconnections, compartmental temperature field with people movement, etc.

Large-scale network

2

3

4

5

1

Figure: Local network

Figure: HVAC system

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Outline

1 Background





6 Conclusions

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Problem formulationThe concerned state-space system model:

x(k + 1) = A(θ)x(k) + B(θ)u(k) + Hf (k)

y(k) = C(θ)x(k) + w(k)

u(k)-measurable input, f (k)-unknown input, y(k)-measurable output. The systemmatrices A(θ), B(θ) and C(θ) are affinely parameterized w.r.t. θ ∈ R

l :

A(θ) = A0+A1θ1+· · ·+Alθl , B(θ) = B0+B1θ1+· · ·+Blθl , C(θ) = C0+C1θ1+· · ·+Clθl .

Example 1

A(θ) =

[θ1 0 θ2

θ2 θ1 00 θ2 θ1

]

, B(θ) =

[θ3

00

]

H =

[010

]

, C(θ) =

[θ4

00

]T

.

1

2 3

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The unknown input f (k) is assumed to be deterministic. No prior knowledgeof f (k) is available except the persistent excitation condition.

In the literature, only the blind SIMO system identification framework canhandle this identification problem.

y1(k) = H1(z)u(k)

y2(k) = H2(z)u(k)

.

.

.

yL(k) = HL(z)u(k)

Since the SIMO model is a decoupled network model, it cannot cope with theidentification of general structured state-space models.

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Outline

1 Background





6 Conclusions

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Identifiability analysis 1 - Full state observation

Given the full state observation, an equivalent state-space model can be written as

x(k + 1) = (A + HQ1)︸︷︷︸

A

x(k) + (B + HQ2)︸︷︷︸

B

u(k) + H [fk − Q1x(k) − Q2u(k)]︸︷︷︸

f (k)

y(k) = Cx(k)

where Q1 ∈ Rr×n and Q2 ∈ R

r×m are ambiguity matrices, and y(k) = y(k) − w(k)denotes the noise-free output.

Without any structural constraints, the system matrices cannot be identified even if thestates can be fully observed. For example, when (A, H) is controllable, the system polescan be arbitrarily assigned.

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Identifiability analysis 1 - Full state observation

Theorem 1. Let P⊥

H = I − H(HT H)−1HT . Suppose that the state sequence x(k) can befully observed, the original identification problem boils down to estimating the parametervector θ from the following modified state-space model

P⊥

H x(k + 1) = P⊥

H A(θ)x(k) + P⊥

H B(θ)u(k)

y(k) = C(θ)x(k).

Example 1 (continue)

[x1(k + 1)x3(k + 1)

]

=

[θ1 0 θ2

0 θ2 θ1

] [x1(k)x2(k)x3(k)

]

+

[θ3

0

]

u(k)

y(k) = θ4x1(k).

The state-equation of the second agent is removed due to the unknown input signal;

however, θ can be estimated because of its duplicates in other state equations. When

H = [1 1 1]T , the system parameters are still identifiable.

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Identifiability analysis 2 - No direct state observation

The past and future data equations are given as

Yp = OXp + TuUp + Tf Fp

Yf = OXf + TuUf + Tf Ff

where

O =

CCA...

CAs−1

, Tu =

0

CB. . .

.... . . 0

CAs−2B · · · CB

, Tf =

0

CH. . .

.... . . 0

CAs−2H · · · CH

,

Xp = [x(1) x(2) · · · x(h)] , Xf = [x(s) x(s + 1) · · · x(s + h − 1)] ,

Yp =

y(1) y(2) · · · y(h)y(2) y(3) · · · y(h + 1)

...... . . . ...

y(s) y(s + 1) · · · y(s + h − 1)

, Yf =

y(s) · · · y(s + h − 1)y(s + 1) · · · y(s + h)

... . . . ...

y(2s − 1) · · · y(2s + h − 2)

.

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Lemma 1. Assume that CH has full column rank and the state-space model described bythe matrix of tuples (A, H, C , 0) is strongly observable , i.e., rank[O Tf ] = n + rank[Tf ].Then, the row subspace of the state space sequence can be estimated as

Row[Xf ] = Row

[Up

Yp

]

∩ Row

[Uf

Yf

]

.

The above lemma indicates that the system state can be estimated up to a similaritytransformation, i.e.,

x(k) = Qx(k)

where x(k) is the state estimate and Q is the similarity transformation matrix.

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Given the state estimate x(k), the state-space model can be written as

Qx(k + 1) = AQx(k) + Bu(k) + Hf (k)

y(k) = CQx(k).

By Theorem 1, the identifiability of θ in the original state-space model is equivalent tothe identifiability of (θ, Q) of the following state-space model

P⊥

H Qx(k + 1) = P⊥

H A(θ)Qx(k) + P⊥

H B(θ)u(k)

y(k) = C(θ)Qx(k).

Theorem 2. Let the low-rank factorization of P⊥

H be given as P⊥

H = UHV TH where

UH ∈ Rn×(n−r) and VH ∈ R

n×(n−r) have full column rank. Then, the above state-spacemodel is identifiable if and only if, for any nonsingular matrices Q∗ ∈ R

n×n andΠ ∈ R

(n−r)×(n−r), the following equations

ΠVTH Q

∗ = VTH Q, ΠV

TH A(θ∗)Q∗ = V

TH A(θ)Q

ΠVTH B(θ∗) = V

TH B(θ), C(θ∗)Q∗ = C(θ)Q

yield that θ = θ∗, Q = Q∗ and Π = I.

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Example 2 The identifiability of the state-space model with the following structuredsystem matrices is considered:

A(θ) =

[θ1 0 θ2

θ2 θ1 00 θ2 θ1

]

, B(θ) =

[100

]

, H =

[010

]

, C(θ) =

[θ3 1 00 0 θ3

]

.

Choosing the matrix VH as V TH =

[1 0 00 0 1

]

, we have the following equation group

[π11 π12

π21 π22

][q∗

1q∗

3

]

=

[q1

q3

][π11 π12

π21 π22

]

,

[θ∗

1 0 θ∗

20 θ∗

2 θ∗

1

] [q∗

1q∗

2q∗

3

]

=

[θ1 0 θ2

0 θ2 θ1

][q1

q2

q3

]

,

[π11 π12

π21 π22

][10

]

=

[10

]

,

[θ∗

3 1 00 0 θ∗

3

][q∗

1q∗

2q∗

3

]

=

[θ3 1 00 0 θ3

] [q1

q2

q3

]

.

For any non-singular matricesΠ and Q∗, it can be verifiedthat Q = Q∗ and θ = θ∗.

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Outline

1 Background





6 Conclusions

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Subspace identification method

Following identifiability analysis, the subspace identification method will be developed bytwo steps:

1 State estimation using the subspace intersection;

2 Parameter estimation by difference-of-convex optimization.

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State estimationBy Lemma 1, the system state can be estimated as follows

Row[Xf ] = Row

[Up

Yp

]

∩ Row

[Uf

Yf

]

.

The subspace intersection can be computed by two steps:

1 Take the SVD as follows

Up

Yp − Wp

Uf

Yf − Wf

=

[U1 U2

][

Σ1

Σ2

] [V T

1

V T2

]

2 Partition U2 into two sub-matrices of the same size U2 =[UT

21 UT22

]T, the

intersection can be computed as

Xf = UT21

[Up

Yp − Wp

]

.

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Lemma 2 If the matrix [O Tf ] is a tall matrix, then the variance of the measurementnoise can be estimated as

σ2w = λmin

[

limh→∞

1

hYpY

Tp − YpU

Tp (UpU

Tp )−1

UpYTp

]

(1)

where λmin(·) represents the least eigenvalue.

Then, an unbiased estimate of the subspace intersection can be computed as follows.

limh→∞

1

h

[Up

Yp

Uf

Yf

][Up

Yp

Uf

Yf

]T

−

0 0 0 0

0 σ2w I 0

[0 0

σ2w I 0

]

0 0 0 0

0

[0 σ2

w I0 0

]

0 σ2w I

=[

U1 U2

] [Σ1

Σ2

][UT

1

UT2

]

.

Partition U2 into U2 =[UT

21 UT22

]T, we have that

Xf = UT21

[Up

Yp

]

= QXf + UT21

[0

Wp

]

︸︷︷︸

estimation error

.

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Parameter estimation

Define the following row sequences

xh = x(s+2 : s+h+1), xh = x(s+1 : s+h), uh = u(s+1 : s+h), yh = y(s+1 : s+h).

Substituting the estimated state into the original state-space model yields

P⊥

H Qxh = P⊥

H A(θ)Qxh + P⊥

H B(θ)uh + η1

yh = C(θ)Qxh + wh + η2,

where η1 and η2 are the state estimation errors that are asymptotically uncorrelated withthe inputs and states.

Step 1. The parameters can be estimated by solving the following optimization problem

minQ,θ

∥∥P⊥

H Qxh − P⊥

H A(θ)Qxh − P⊥

H B(θ)uh

∥∥

2

F+ ‖yh − C(θ)Qxh‖2

F .

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Step 2. The bilinear estimation problem is equivalent to a rank-constrained optimizationproblem

minQi ,A,C,θ,Γ

∥∥P⊥

H Qxh − P⊥

H Axh − P⊥

H B(θ)uh

∥∥

2

F+ ‖yh − Cxh‖2

F

s.t. A = A0Q +

l∑

i=1

Ai Qi , C = C0Q +

l∑

i=1

Ci Qi

Γ =

[1 θ1 · · · θl

vec(Q) vec(Q1) · · · vec(Ql )

]

rank[Γ] = 1 (‖Γ‖∗ − ‖Γ‖2 = 0)

Step 3. By treating the nonnegative constraint ‖Γ‖∗ − ‖Γ‖2 as a penalty, we can obtainthat

minQi ,A,C,θ,Γ

∥∥P

⊥

H Qxh − P⊥

H Axh − P⊥

H B(θ)uh

∥∥

2

F+ ‖yh − Cxh‖

2F + λ (‖Γ‖∗ − ‖Γ‖2)

s.t. A = A0Q +

l∑

i=1

Ai Qi , C = C0Q +

l∑

i=1

Ci Qi

Γ =

[1 θ1 · · · θl

vec(Q) vec(Q1) · · · vec(Ql )

]

This optimization problem is solved by the sequential convex programming approach.21 / 29

Outline

1 Background





6 Conclusions

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Blind identification of an SIMO modelConsider the following SIMO model y(k) =

∑L

i=1hiu(k − i) + w(k) or

x(k + 1) = Ax(k) + Hf (k), y(k) = Cx(k) + w(k)

where

A =

01 0

. . .. . .

1 0

, H =

10...

0

, C =[

h1 h2 · · · hL

].

The data equation has the following extended observability matrix and convolution matrix

O =

h1 · · · hL

0. . .

......

. . . h1

0 · · · 0... · · ·

...

0 · · · 0

, Tf =

0hL 0...

. . .. . .

h1 · · · hL 0...

. . .. . .

. . . 00 · · · h1 · · · hL

.

It can be seen that [O Tf ] is exactly a convolution matrix of the FIR model.

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10 20 30 40 50 60 70 80 90 100 110

SNR (dB)

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

NE

Es

Subspace fitting method

The proposed method

Figure: Comparison between the subspacefitting method and the proposed method.

200 400 600 800 1000

Data length

0

2

4

6

8

10

12

14

16

18

NE

Es

10-3

Figure: Performance of the proposedmethod against the data length.

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Structured state-space model

Consider Example 2 with the system matrices

A(θ) =

[θ1 0 θ2

θ2 θ1 00 θ2 θ1

]

, B(θ) =

[100

]

, H =

[010

]

, C(θ) =

[θ3 1 00 0 θ3

]

and the parameter vector θ =[

0.5 0.3 1]T

.

It has been verified that this structured state-space model is identifiable in the presence

of the unknown input signal.

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10 20 30 40 50 60 70 80 90 100 110

SNR (dB)

10-8

10-6

10-4

10-2

100

102

104

106

NE

Es

Nuclear norm method

Gauss-Newton method

The proposed method

Figure: Performance comparison among theGauss-Newton method, nuclear-normmethod and the proposed method.

200 400 600 800 1000

Data length

0

2

4

6

8

10

12

14

16

18

NE

Es

10-3

Figure: The recovered unknown input signalat SNR=30dB.

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1 Background





6 Conclusions

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Conclusions

Conclusions

1 Identifiability conditions for the structured state-space models

2 Subspace-based state estimation with noise compensation

3 Difference-of-convex programming approach for the bilinear estimation

Future work

Simultaneous system structure reconstruction and the system parameter estimation using(partially) measurable input-output data.

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THANK YOU!

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