Closed-loop MOESP subspace model identification ...

Delft University of Technology Delft Center for Systems and Control

Technical report 10-015

Closed-loop MOESP subspace model identification with parametrisable disturbances Gijs van der Veen, Jan-Willem van Wingerden, Michel Verhaegen

c Delft Center for Systems and Control Copyright All rights reserved

Delft University of Technology Mekelweg 2, 2628 CD Delft The Netherlands Phone: +31-15-27 85623 Fax: +31-15-27 85602 http://www.dcsc.tudelft.nl

Closed-loop MOESP subspace model identification with parametrisable disturbances Gijs van der Veen, Jan-Willem van Wingerden, Michel Verhaegen Delft Center for Systems and Control, Delft University of Technology, Delft, The Netherlands

August 2010, (Last modifications: September 21, 2011) Abstract A new subspace identification method for systems operating either in open-loop or in closed-loop is presented. The method obtains an estimate of the innovation sequence by performing an RQ-factorization of the measurement data, thereby avoiding explicitly solving a least-squares problem. In a second step, the estimated innovation sequence is used to perform ordinary MOESP (Verhaegen, 1994) to find the system matrices up to a similarity transformation. The closed-loop identification algorithm is extended to cases where certain inputs are not persistently exciting, which occurs when those inputs have a narrowband character. The method is illustrated by applying it to several examples.

1 Introduction Subspace identification methods have proven to be valuable tools in the field of system identification and have undergone significant developments in the past two decades. More recently, the interest has shifted to closed-loop extensions of the standard subspace identification tools (Chou and Verhaegen, 1997; Qin and Ljung, 2003). In (Oku and Fujii, 2004), a twostage method is used that is similar to the method described in (Van den Hof et al., 1992) for prediction-error methods where the sensitivity function is first identified. In (Gilson and Merc`ere, 2006) a modification is made to the PO-MOESP scheme to accommodate the correlation between the noise and the instrumental variables. In this paper a subspace identification method (CL-MOESP) is presented that stays close to the ordinary MOESP algorithm (Verhaegen, 1994). In a first step, the innovation sequence is estimated, which is then treated as a known sequence in subsequent steps. The innovation estimation step has similarities with certain steps taken in (Qin and Ljung, 2003) and (Chiuso, 2007), but the subsequent steps differ. The closed-loop identification method is further extended to treat the case in which one or more disturbance inputs are present that can be parametrised in terms of basis functions related to a known signal. One case of interest where this occurs is when data is captured from operational wind turbines (van der Veen et al., 2010; van Wingerden et al., 2010). Such data often contains strong periodic components at frequencies that are harmonics of the rotor frequency. These effects can be dealt with in a straightforward way by generating additional input signals (van Baars and Bongers, 1994). In some cases, adding such signals leads to an ill-conditioned identification problem, especially when they have a narrowband character. A technique to deal with such situations is presented in the second part of this paper. The paper starts with an introduction to the identification problem. In section 3, the main steps of the CL-MOESP algorithm are described. In section 4 the methods are described that allow treatment of inputs that may not satisfy the persistence of excitation condition. The paper concludes with a demonstration of the algorithm applied to both a theoretical closed-loop identification problem and an experimental problem of identifying a “smart” rotor system. The paper ends with some conclusions and recommendations for future work.

2 Problem statement We wish to identify a dynamic model of a system on the basis of input and output measurements obtained from the system operating in open-loop or closed-loop. Thus, we have a set of measurements {uk , yk }N k=1 and seek a state-space model. The system to be identified is assumed to admit an innovation representation given by: xk+1 = Axk + Buk + Bd dk + Kek ,

(1a)

yk = Cxk + Duk + Dd dk + ek ,

(1b)

with A ∈ Rn×n , B ∈ Rn×m , Bd ∈ Rn×md , K ∈ Rn×l , C ∈ Rl×n and D ∈ Rl×m . The vectors xk ∈ Rn , uk ∈ Rm , dk ∈ Rmd , yk ∈ Rl and ek ∈ Rl are the state sequence, input, disturbance, output and innovation, respectively. K is the Kalman gain. The innovation sequence ek is an ergodic zero-mean white noise sequence with covariance matrix E{ej e⊤ k } = W δjk , with

3

W ≻ 0. It is assumed that the eigenvalues of A − KC are strictly inside the unit circle, which is equivalent to the natural assumption 1 that the model to be identified is reachable and observable. The pair (A, C) is observable and the pair (A, [B KW 2 ]) is controllable. Note that the description allows for a disturbance input dk . It is assumed that this input can be linearly P parametrised in terms of basis functions that are related to a known signal, e.g. dk = i αi φi (gk ), where gk is a known signal and φi (·) are the basis functions. Alternatively, ek may be eliminated from the first equation to yield a system description in one-step-ahead predictor form: ˜ k + Bu ˜ k +B ˜d dk + Kyk , xk+1 = Ax (2a) yk = Cxk + Duk + Dd dk + ek ,

(2b)

˜ ≡ B − KD have been introduced for convenience of notation, B˜d is defined likewise. where A˜ ≡ A − KC and B

3 Closed-loop subspace identification One of the main reasons that open-loop subspace methods that rely on instrumental variables, such as PO-MOESP (Verhaegen and Verdult, 2007) and N4SID (Overschee and Moor, 1996), deliver biased estimates in a closed-loop setting is the underlying assumption that the input signal uk is uncorrelated with the past noise process ek . In a closed-loop situation, however, it is clearly seen that this condition is violated: E{uk e⊤ j } 6= 0 for j < k. For this reason, a first step in the algorithm presented here is to obtain an estimate of the innovation sequence as proposed in (Qin and Ljung, 2003), so that it is completely defined in the further steps of the subspace identification procedure. The steps taken to estimate the parameters of the innovation model (1) are to first estimate the innovation sequence, then estimate the extended observability matrix to find A and C, and finally to estimate B, D and K in a least-squares problem.

3.1 Deriving the data equations (f )

Before deriving the data equations for subspace identification, we will introduce some notation. The stacked vector Uk defined as  ⊤ (f ) Uk = u⊤ , u⊤ · · · , u⊤ k, k+1 , k+f −1 (f )

(f )

is

(f )

where f signifies the future window size. Similarly, vectors Yk , Dk and Ek exist. We also define block-Toeplitz matrices ˜ B, ˜ D) pertaining to the innovation (1) and predictor (2) models respectively: H(B, D) and H(   D 0 0 ··· 0  CB D 0 ··· 0      CAB CB D · · · 0 H(B, D) =  ,   .. . . . . . .   . . . . CAf −2 B CAf −3 B · · · CB D   D 0 0 ··· 0 ˜  CB D 0 ··· 0    ˜ ˜  C A˜B C B D · · · 0  ˜ ˜ H(B, D) =  .   .. .. .. ..   . . . . ˜ · · · CB ˜ D ˜ C A˜f −3 B C A˜f −2 B ˜ B ˜d , Dd ), H(K, ˜ Likewise, we define matrices H( 0), H(Bd , Dd ) and H(K, 0). We also define extended controllability matrices Ku , Kd and Ky :   ˜ A˜f −2 B, ˜ ··· , B ˜ , Ku = A˜f −1 B,   f −1 ˜d , · · · , B ˜d , ˜d , A˜f −2 B Kd = A˜ B   Ky = A˜f −1 K, A˜f −2 K, · · · , K .

Finally, the extended observability matrices are defined:  C  ˜= Γ



˜ CA

.. .

˜f −1 CA

,

Γ=

4

C CA

.. . CA

f −1



.

Starting from some initial state xk , the state equation can be propagated  ahead in time, resulting in the expression: (f ) U   k  xk+f = A˜f xk + Ku Kd Ky  Dk(f )  . (3) (f ) Yk

Based on (3) and the output equation, future outputs can be written as: yk+f = Cxk+f + Duk+f + Dd dk+f + ek+f .

When f subsequent future outputs yk to yk+f −1 are gathered in a vector, one obtains: (f ) ˜ k + H( ˜ B, ˜ D)U (f ) + H( ˜ B ˜d , Dd )D(f ) + Y = Γx k

k

k

(f ) (f ) ˜ H(K, 0)Yk + Ek .

To be able to solve for the unknowns in a least-squares sense, the data columns are augmented with shifted versions. To this end, past and future Hankel matrices are defined as follows, assuming that all samples at our disposal are used:  y1 y2 ··· y   yp+1 yp+2 ··· y  y2

 Yp =  .. .

y3

.. .

N −f −p+1

··· yN −f −p+2

..

.

yp yp+1 ...

.. .

yN −f

yp+2

   , Yf =  .. .

yp+3

.. .

N −f +1

··· yN −f +2

..

.

yp+f yp+f +1 ...

.. .

yN

 .

The subscripts p and f denote the past and future horizon respectively. Note that the past and future Hankel matrices have block row dimensions p and f respectively. In exactly the same way Up , Uf and Ef are defined. The data equation incorporating all measurement data can now be written as: ˜ f + H( ˜ B, ˜ D)Uf + H( ˜ B ˜d , Dd )Df + H(K, ˜ Yf = ΓX 0)Yf + Ef ,   where Xf = xp+1 · · · xN −f +1 represents the state sequence. Note that this state sequence can be approximated using (3), under the assumption that A˜p Xp ≈ 0 (Knudsen, 2001):     Up Xf ≈ Ku Kd Ky  Dp  = KZp . Yp

Here, KZp has been introduced as a short-hand notation with Zp = [ Up⊤ , Dp⊤ , Yp⊤ ]⊤ containing past input-output data. By the assumption that A˜ has all its eigenvalues inside the unit disc, the term A˜p Xp can be made arbitrarily small by choosing p sufficiently large. The complete data equation may now be written as : ˜ ˜ ˜ ˜ ˜ Yf = ΓKZ p + H(B, D)Uf + H(Bd , Dd )Df + ˜ H(K, 0)Yf + Ef .

(4)

Also note that when all data points are used, the samples k = (1, N − f ) are assigned to the past Hankel matrices and the samples k = (p + 1, f ) are assigned to the future Hankel matrices.

3.2 Estimation of the innovation sequence Upon taking a closer look at the data equation (4), it can be seen that the first block row Yf (1:l, :) does not depend on ˜ Yf on the right-hand-side of the equation (based on the structure of H(K, 0) having a zero block-diagonal). Therefore, considering this first block row, it is possible to estimate the first row of the innovation Hankel matrix Ef (1:l, :). Yf (1:l, :) = CKZp + DUf (1:l, :) + Dd Df (1:l, :) + Ef (1:l, :). The first block-row of the equation can be solved for the unknown parameters in a least squares sense, i.e.

 

Zp

 

Yf (1:l, :) − CK D Dd  Uf (1:l, :)  , min (5)

[ CK D Dd ] Df (1:l, :) F   ˆf (1:l, :) is the least-squares residual. However, explicitly determining the parameters CK D Dd will be such that E avoided which is more attractive from a computational point of view. Instead, an orthogonal projection will be applied via the RQ-factorisation (Golub and Van Loan, 1996) to find the residual, i.e. that part of Yf (1:l, :) which is orthogonal to the row space of [ Zp⊤ Uf (1:l,:)⊤ Df (1:l,:)⊤ ]⊤ . Performing an RQ-factorization of the available data one obtains:     Zp #" # " (1) (1)   Uf (1:l, :)   Q1 0  = R11  , (1) (1) (1)   Df (1:l, :) R21 R22 Q2 Yf (1:l, :)

5

ˆf (1:l, :) = R(1) Q(1) . From this estimated from which it can be derived, using the orthogonality of the rows of Q, that E 22 2 N Hankel matrix the estimated noise sequence {ˆ e}k=p+1 can be reconstructed. Note that the least-squares problem (5) is also considered as a first step in predictor based subspace identification PBSID (Chiuso, 2007), where a high order ARX model is fitted to the data. However instead of explicitly estimating the parameters in the least squares problem (5), only the residual sequence Ef (1 : l, :) is estimated. From there on, the proposed algorithm aims at estimating the column space of the extended observability matrix instead of estimating the state sequence of (a set of) Kalman filters as in N4SID approaches (Overschee and Moor, 1996). Given the estimate of the matrix Ef , we return to the innovation representation of the data equation based on (1), which reads: Yf = ΓXp + H(B, D)Uf + H(Bd , Dd )Df + H(K, 0)Ef .

(6)

Note that the data equation differs in a subtle way from (4), since it contains the parameters of the innovation form (e.g. A ˜ The estimate Ef found in this section is now considered as and B) as opposed to the predictor parameters (e.g. A˜ and B). a known input to the innovation system in (6) and therefore the ordinary MOESP (Multivariable Output-Error State-sPace) identification scheme for deterministic systems may be used. A summary of ordinary MOESP is given in the next subsection.

3.3 Estimating the matrices A and C ⊤ ⊤ −1 In (6) the matrices Uf , Df and Ef are known. Therefore, an orthogonal projection matrix Π⊥ Zf , with Zf = I − Zf (Zf Zf )

Zf = [ Uf⊤ , Df⊤ , Ef⊤ ]⊤ , can be constructed that removes these matrices from the right hand side of the equation resulting in: ⊥ Yf Π ⊥ Zf = ΓXp ΠZf . In practice, this projection may be obtained by performing an RQ-factorization of the input and output data, which is numerically much more efficient than evaluating the large projection matrix: #" #   " (2) (2) Q1 Zf R11 0 . = (2) (2) (2) Yf R21 R22 Q2 This equation projects Yf onto the null space of the joint input and noise Hankel matrix Zf . Now, we can equivalently write: (2)

(2)

⊥ Yf Π ⊥ Zf = ΓXp ΠZf = R22 Q2 .

If the input and noise sequences are persistently exciting of at least order f m, f md and f l respectively for the input, disturbance and innovation signals (Verhaegen and Verdult, 2007), the following holds, since the column space of the observability matrix is preserved in Yf Π⊥ Zf after projection: (2)

range(Γ) = range(Yf Π⊥ Zf ) = range(R22 ). (2)

Thus, the column-space of R22 serves as a basis for the column space of the extended observability matrix Γ. Performing a (2) singular value decomposition of R22 gives: (2) R22 = Un Σn Vn⊤ , where n is the number of dominant singular values and also the order of the underlying system including the noise model. The columns of Un provide a basis for Γ. A gap between successive singular values will often indicate the order of the system (see (Verhaegen, 1993) for more details). Estimates of the matrices A and C can subsequently be obtained from Un . Given the structure of Γ, the C-matrix is found as the first l rows of Un . A can be found as the solution to the overdetermined problem Un (1:(f − 1)l, :)A = Un (l + 1:f l, :) (using M ATLAB notation). The matrices B, D and K can be computed in a second step by solving a least-squares problem as shown in the next subsection.

3.4 Estimating B, D, K and the initial state Based on the innovation form of the system, the output at time k can be written as: yk = CAk x0 +

k−1 X

CAk−τ −1 (Buτ + Bd dτ + Keτ ) +

τ =0

Duk + Dd dk + ek .

Applying the vectorisation operator and exploiting the properties of the Kronecker product, this can be rewritten as:  x0  vec(B)

yk = [ Φxk0 |

B

D

d ΦK ΦD Φ d ΦB k Φk k k k

{z

Φk

 vec(Bd )  ]  vec(K)  +ek , } vec(D) vec(Dd )

|

6

{z Θ

}

(7)

where we have defined:  k−τ −1 u⊤ , τ ⊗ CA P  k−1 ⊤ K k−τ −1 Φk = , τ =0 eτ ⊗ CA
D Φk d = d ⊤ ⊗ I . k

Φxk0 = CAk−1 ,  P B k−1 ⊤ k−τ −1 , Φk d = τ =0 dτ ⊗ CA
⊤ ΦD = u ⊗ I , k k

ΦB k =

P

k−1 τ =0

This is a linear expression in the unknown elements of x0 , B, Bd , D, Dd and K, which can be solved for the parameters in a least-squares sense with the available data set {uk , yk , ek }N k=p+1 . For this purpose, define Φk and Θ as in (12). Then the least squares problem can be stated as: ˆ = arg min Θ Θ

N X

kyk − Φk Θk22 = arg min kY − ΦΘk22 , Θ

k=1

which can be solved efficiently using a QR-factorization (Golub and Van Loan, 1996) as follows: # " h i R(3) R(3) (3) (3) 11 12 ˆ = (R(3) )−1 R(3) . [Φ Y ] = Q1 Q2 → Θ (3) 11 12 0 R22 It must be noted that, although the described way to find B, Bd , D, Dd , K and the initial state is conceptually simple, it is computationally prohibitive due to the Kronecker products involved. Much more efficient procedures exist to find the matrix Φ which avoid evaluation of Kronecker products for each k. Such procedures are detailed in (Verhaegen and Varga, 1994) and section 3.6. The approach presented here must be slightly modified in case the system is unstable, see (Chou and Verhaegen, 1997).

3.5 Estimating B, D and K when A is unstable (Chou and Verhaegen, 1997) ¯ be any matrix such that A − KC ¯ is For simplicity we consider a situation without periodic disturbances, dk = 0. Let K ¯ k to the innovation model state equation we obtain: stable1 . Adding and substracting Ky ¯ ¯ ¯ ¯ xk+1 = (A − KC)x k + (B − KD)uk + (K − K)ek + Kyk , yk = Cxk + Duk + ek .

(8) (9)

We also define the auxiliary state-space model: ¯ ¯ ηk+1 = (A − KC)η k + Kyk ,

(10)

ξk = Cηk .

(11)

It can be shown that the following equality holds: ¯ k x0 + yk − ξk = C(A − KC)

k−1 X τ =0

  ¯ k−τ −1 (B − KD)u ¯ ¯ C(A − KC) τ + (K − K)eτ +

Duk + ek .

Applying the vectorisation operator and exploiting the properties of the Kronecker product (Brewer, 1978), this can be rewritten as:   x0 ¯    vec(B − KD)    +ek , yk − ξk = Φxk0 ΦB (12) ΦD ΦK k k k ¯   vec(K − K) | {z } Φk vec(D) | {z } Θ

with the earlier definitions. Following this output identity, the matrices B and K can be recovered from the least-squares solution in a straightforward manner. 1 The

¯ is introduced to avoid confusion with K of the innovation model. notation K

7

3.6 An efficient way to find B, D, K and the initial state In this section, an alternative way to estimate the unknown matrices and the initial state is described that exploits the knowledge of the pair (A, C) to simulate subsystems of the complete system. The approach described here is partly based on (Verhaegen and Varga, 1994). The state equation of the system can be written as xk+1 = Axk + Buk + Kek = Axk + eˆ1 uk (1)B11 + . . . + eˆn uk (m)Bnm + eˆ1 ek (1)K11 + . . . + eˆn ek (l)Knl . In this equation, eˆi is the ith basis vector in cartesian space and uk (j) is the j th element of uk . Based on this representation the following subsystems can be defined, each of which defines the response of the output due to an unknown element of B or K, assuming a zero initial state: ( ij xk+1 = Axij ˆi uk (j) k +e for B : with i = 1 . . . n, j = 1 . . . m B ij ij yk = Cxk , ( ij xk+1 = Axij ˆi ek (j) k +e for K : with i = 1 . . . n, j = 1 . . . l K ij ij yk = Cxk , The same procedure can be followed for the output equation, where the response due to each element of the D matrix is found, yk = Duk = eˆ1 uk (1)d11 + . . . + eˆl uk (m)dlm , so that the following subsystems can be recognised:  for D : D ykij = eˆi uk (j),

with i = 1 . . . l, j = 1 . . . m

The relation between the output of the subsystems and the true output at time k is is:   B11  .   ..        k−1 B 12 B n1 B 11 yk = CA · · ·  Bn1  + | yk ··· yk | {z } x0 + | yk  {z }  B12  O(k) YB (k)  .  ..   K11  .   ..     K 11   K 12 K n1 · · ·  Kn1  + D yk11 · · · D ykl1 | | yk ··· yk yk  {z } | {z  K12  | YK (k) YD (k)  .  ..



D 12 yk

    ···  }  

D11 .. . Dl1 D12 .. .



     + ek   

Note that the influence of the (unknown) inital state also appears explicitly at this point. Since the pair (A, C) is known, the subsystems for the elements of B, D and K can be simulated to obtain the required data. It is now possible to estimate x0 and the vectorised matrices B, D and K by solving a linear least-squares problem. A system without the feedthrough term can be estimated by leaving out the part corresponding to the D-matrix. Collecting the outputs for k = [1, N ] the least-squares problem can be formulated:       x0 y1 e1   vec B   .   .     (13) +  ..   ..  = O YB YK YD  vec K  yN e N vec D

This problem can be solved efficiently with the aid of an orthogonal transformation, e.g. a QR factorisation (Golub and Van Loan, 1996).

3.7 Finding a stabilising Kalman gain K In general, there is no guarantee that the matrix K found in the previous section results in a stable one-step-ahead predictor with all eigenvalues of A−KC strictly inside the unit circle. An alternative way to obtain K is by solving a discrete algebraic Riccati equation (DARE) to find the steady-state Kalman gain based on the pair (A, C) and sample estimates of the process ˆ = PN wi wi⊤ , R ˆ = PN vi vi⊤ and Sˆ = PN wi vi⊤ . An estimate of and measurement noise covariance matrices, Q i=1 i=1 i=1

8

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/(.'*&0&12*&3') 4,31.(+,.

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Figure 1: The proposed way of dealing with periodic disturbances: The measured identification data yk is perturbed by unknown disturbances dk that can be related to a known signal ψk ; the 1P and 2P periodic signals are added to the set of input signals uk used for identification in order to suppress the effect of the disturbances on the quality of the identified model of uk → yk .

the process noise sequence can be simply obtained by forming the product of the estimated innovation sequence and the estimated Kalman gain: ˆf (1: l, :)K ˆ ⊤. W1:l,: = E K is then found as ˆ + CP C ⊤ )−1 , K = (Sˆ + AP C ⊤ )(R where P is the unique positive-definite solution to the DARE ˆ − (Sˆ + AP C ⊤ )(CP C ⊤ + R) ˆ −1 (Sˆ + AP C ⊤ )⊤ . P = AP A⊤ + Q

4 Dealing with low rank input Hankel matrices In some cases, strong periodic disturbance inputs may be present that can be related to a measured signal (e.g. the rotor azimuth in the case of a wind turbine). Such strong periodic components fail to match the standard hypothesis in system identification that measurements are corrupted with a stochastic noise component. Since output measurements may be highly influenced by the periodic disturbances, the standard identification procedures are likely to search for a causal relation between the applied inputs and the outputs which is not present at the frequencies of the periodic disturbances. In many cases this will lead to a poor description of the input-to-output behaviour of the system (van der Veen et al., 2010). If the output measurements are to a large extent corrupted with periodic signals of known frequencies, it is possible to construct virtual input signals with corresponding frequencies that are able to account for periodic components in the outputs (van Baars et al., 1993). The operation of such signals can be explained for the example of a wind turbine: the outputs are affected by periodic signals of unknown amplitude and phase, but which are directly correlated to the rotor azimuth ψk and higher harmonics. Such a signal can be constructed from a linear combination of a cosine and sine function with unit amplitude and zero phase: A sin (ψk + φ) = α sin (ψk ) + β cos (ψk ). Thus, input signals can be generated that account for the periodic components in the output. As shown in Figure 1, these constructed signals can be added as inputs to the identification procedure. Adding such inputs, however, may require some modifications to the subspace algorithm to avoid solving a rank-deficient least-squares problem. These modifications are treated in more detail in the following subsections.

4.1 The subspace projection step If one or more of the inputs do not fulfill the persistency of excitation condition, the standard rank condition in MOESP is no longer satisfied. It will be assumed here that there is at least one input that is persistently exciting of sufficiently high

9

order for the system and that the inputs which do and do not satisfy this condition have been separated into two input Hankel matrices, Ufp and Ufnp . Doing so, the Hankel matrix of the first set of inputs can be used to perform the projection without further considerations. The projection based on the second set of inputs, however, needs to be examined more closely. The second set of inputs is such, that the Hankel matrix constructed from them is rank deficient. The following SVD is performed:     V1⊤ np Uf = U Σr 0 , (14) V2⊤

such that V1 has full rank r Assuming the following holds:   Xp rank = n + r, V1⊤

(15)

it holds that 

Xp Xp⊤ V1⊤ Xp⊤

 X p V1 ≻ 0. V1⊤ V1

(16)

Applying the Schur complement, it then holds that   rank Xp Xp⊤ − Xp V1 (V1⊤ V1 )−1 V1⊤ Xp⊤ = n.

(17)

From this, it can be deduced that

  ⊥ p np = n, rank Yf Π⊥ U ΠU f

(18)

f

⊥ p np : which implies that the column space of the extended observability matrix equals the column space of Yf Π⊥ U ΠU f





⊥ p np . range (Os ) = range Yf Π⊥ U ΠU f

f

f

(19)

The main underlying assumption is that the condition in (15) holds, which effectively means that the state sequence has full rank and it does not have rows that are in the null space of (Ufnp )⊥ . Thus, it must be ensured that there are inputs to the system that excite all states and are sufficiently persistently exciting. If that is the case, adding inputs that are not persistently exciting does not cause the rank conditions to be violated, so that the extended observability matrix can still be found.

4.2 The step of estimating B, D, K and the initial state B

D

Inputs that are not persistently exciting of sufficiently high order may result in matrices Φk d or Φk d in (12) that have linearly dependent columns, causing the least-squares problem to become rank-deficient. This occurs, for example, in the case of periodic inputs. A minimum-norm least-squares solution will still minimise the least-squares residual, but possibly at the cost of a very sensitive solution with poorly scaled numerical values. To avoid this problem, the proposed solution is to compress the set of columns of Φk that correspond to input dk using an SVD when necessary. For example:    Σν  ΦBd = Uν U ⊥ V ⊤. (20) 0

This step results in a full rank compressed matrix Uν containing an orthogonal basis for ΦBd . ΦBd in the least-squares problem can then be replaced with Uν and the full-rank least-squares problem can be solved for a compressed parameter vector. After a solution has been found, the parameters of Bd may be found from (20).

5 Examples and results 5.1 A theoretical example To demonstrate the effectiveness of the presented method, we apply it to a closed-loop identification problem adapted from (Van den Hof et al., 1992; Oku and Fujii, 2004). The closed-loop system shown in Figure 2 is simulated with G=

1 1 − 1.6q −1 + 0.89q −2

C = q −1 − 0.8q −2 H=

1 − 1.56q −1 + 1.045q −2 − 0.3338q −3 . 1 − 2.35q −1 + 2.09q −2 − 0.6675q −3

10

e H

r

u −

y

G

C Figure 2: The closed-loop identification setting from (Van den Hof et al., 1992).

The noise input e and reference r are both zero-mean white noise signals with unit variance. The input u and output y are measured for identification of system G. For this example, past and future window sizes of 10 are chosen and a model of 3rd order (note that G and H have two poles in common) is identified using 1000 samples. Figure 3 shows the pole locations of the identified models for 25 independent realisations using PO-MOESP, PBSIDopt (Chiuso, 2007) and the proposed CLMOESP algorithm. The figure shows that the poles are distributed in a similar way around the true pole location compared to PBSID. The figure also shows that PO-MOESP yields biased estimates in this closed-loop setting.

0.65 0.6

Im{λ}

0.55 0.5 0.45 0.4 0.35 0.3 0.6

0.7

0.8 Re{λ}

0.9

1

Figure 3: Pole locations for the true and identified models using three different methods. 25 realisations are shown.

5.2 An experimental example In a second example, the method is applied to measurements taken from a scale model of a wind turbine (Hulskamp et al., 2010) at Delft University of Technology. The scale model has two “smart” rotor blades, where each of the blades is equipped with piezoelectric trailing edge flaps. These devices are used to alleviate the blade loads by modifying the the local aerodynamic loads. The CL-MOESP identification procedure was used to identify linear state-space models of the system (van der Veen et al., 2010). The case under consideration concerned the identification of the dynamics from the trailing edge flap actuators on each of the two blades to the strain measurements in the root of the blades. The strains are measured using piezoelectric macro fiber composite (MFC) strain gauges. Each of the trailing edge flaps was subjected to an independent input sequence. In an experiment, 10000 samples of I/O data were obtained at a rate of 100 Hz. As input signals two pseudorandom binary sequences were used with amplitudes of 400 V. To determine the problematic periodic components, the power spectra of the output signals were analysed. These showed distinct peaks at 1P and 3P frequencies. Figure 4 shows how the periodic components affect the output measurements: the excitation capability of the actuators is limited and therefore the strong components at 1P and 3P are still strongy represented in the output measurements. In Figure 5, the results of an identification experiment are shown. The strong effect of the periodic signals on the output measurements

11

Power [dB2 /Hz]

-50

-100

-150

0

10

Figure 4: Output spectra for measurements with ( nance of the periodic effects at 1P and 3P.

20 30 Frequency [Hz]

) and without (

40

50

) an excitation signal present, showing the domi-

forces the identification procedure to identify incorrect dynamics specifically at the 1P and 3P frequencies to account for the presence of these frequencies in the output signal. With the addition of periodic signals as described before, these incorrect dynamics are no longer present. In Figure 6 the bode magnitude diagram of the complete 12th order model is shown. The identified models have been used successfully in a model based feedforward-feedback control design which was applied to the “smart” rotor model (van Wingerden et al., 2010).

MFC blade 1 [V]

From Flap blade 1 [V]

10−2

10−3

101 Frequency [Hz]

Figure 5: Comparison of two linear models from one the trailing edge actuators to a strain measurements at the blade root showing the mismatch at 1P resulting from a standard identification procedure. The grey line ( ) corresponds to the ) corresponds to identified model obtained with identified model without adding periodic input signals, the black line ( added periodic signals. The results shown correspond to the case ω = 430 rpm and v = 10 m/s.

5.2.1 Model validation The quality of the identified models can be assessed in several ways. There is no way to obtain an exact linearised reference model in this case. Therefore, the identified frequency response functions are compared to the empirical transfer function ˆ i→j (ejωk ) based on the FFTs of the input-output data. The estimates are shown in Figure 6 together with the estimates H identified models. On the whole, the results are quite accurate. It is observed that for some cases there are slight differences in the low-frequency behaviour. As a means of cross-validation, the dataset was split up into a 2/3 part for identification and a a 1/3 part for validation. As a quality measure, the variance-accounted-for (VAF) was used, which gives a measure of how well the variability of the output signal is predicted by the linear model and is expressed as ! PN ˆj )2 j=1 (yj − y × 100%, VAF = 1 − PN 2 j=1 yj

12

MFC blade 1 [V]

From Flap blade 1 [V]

10−2

10−2

10−3

10−3

10−4

MFC blade 2 [V]

From Flap blade 2 [V]

10−4

101

10−2

10−2

10−3

10−3

10−4

10−4

101 Frequency [Hz]

101

101 Frequency [Hz]

Figure 6: Comparison of the estimated linear model with spectral estimates of the transfer function from the trailing edge actuators to strain measurements at the blade root. The conditions correspond to ω = 370 rpm and v = 7 m/s. The gray line corresponds to the models identified without addition of periodic signals.

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where yˆ is the output predicted by the identified model. The VAF values were around 95% for both the identification and validation data sets, indicating that a good quality model has been found and that the model does not suffer from overfitting, in which the model is fitted to the experiment noise. A second test for model quality can be carried out by examining the autocorrelation spectra of the prediction errors ǫj = yj − yˆj . If the model has truly fitted the input-output data accurately, the prediction error signals should be white noise signals, indicating that all correlation has been removed from the data and all information has been extracted from the signal. In Fig. 7 these spectra are shown together with the 99% confidence bounds and it can be seen that the residual signals are almost uncorrelated. An examination of the spectral content reveals that the correlation still present in the residuals is dominated by some unmodelled periodic components at higher harmonics of the rotor speed. 1.2

Rǫǫ (τ )/Rǫǫ (0)

1 0.8 0.6 0.4 0.2 0 -0.2

-20

-10

0 τ

10

20

Figure 7: Normalised autocorrelation spectra of the output residual signals between the predicted and true output signals. The concentrated peak at τ = 0 indicates that the signals are almost white.

6 Conclusions In this paper a method for the identification of MIMO state-space models from closed-loop operational data was described. The method is based on the ordinary MOESP scheme for open-loop identification. Additionally, methods have been described to deal with the case that some of the input signals are of a narrowband character, which may occur in several practical situations as for example in data obtained from operational wind turbines. The applicability of the method has been demonstrated in both theoretical and experimental contexts, showing that the identification technique gives useful models for subsequent controller design. It is hoped that the identification techniques presented here can be extended to the case of linear time-varying (gain-scheduled) systems to be able to model wind turbine dynamics over a wider operational range.

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