A modified method for closed-loop identification of

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A Modified Method for Closed-Loop Identification of Transfer Function Models Wei Xing Zheng

Abstract—Substantial revisions on the newly proposed bias correction based method are made in the framework of indirect identification of a linear (possibly unstable) plant operating in closed loop with a low-order stabilizing controller. By making a new formulation of a least-squares estimate of an intermediate parameter vector purposely introduced, the modified algorithm is able to achieve a direct yet unbiased closed-loop system estimate in the presence of misspecified noise model. With no identification of a high-order augmented closed-loop system, the computational complexity of the algorithm is significantly reduced. Simulations of identifying an open-loop unstable plant illustrate the promising performance of the modified algorithm in low signal-to-noise ratio environments. Index Terms—Algorithm implementation, closed-loop systems, identification, least-squares method, parameter estimation.

I. INTRODUCTION The goal of closed-loop identification is to acquire good transfer function models of the open-loop plant in the face of the feedback. In recent years, there has been tremendous renewed interest in closed-loop identification, and, in particular, its importance has been recognized from the viewpoint of control systems design (see, e.g., [3], [11]). It is shown that for control systems design, closed-loop identification of the plant is often of far great value than its open-loop identification. There are many examples where a plant estimate obtained through closed-loop identification provides good and effective control performance. Furthermore, re-identification of the plant in closed loop may make it much convenient to re-tune the controller for the purpose of controller maintenance [4]. This paper considers closed-loop identification by use of the indirect approach [5], [8]. The distinctive feature of this frequently used approach is that the knowledge of the regulator in the loop is required when an estimate of the open-loop plant is being calculated after identification of the corresponding closed-loop system. In practice, it is usually the case that the disturbance acting on the system can not be modeled exactly. Despite its simplicity, the effectiveness of the conventional least-squares (LS) method depends completely upon the correctness of the noise model, thus severely limiting its applicability. Several indirect identification algorithms that can give unbiased parameter estimates irrespective of noise dynamics are available, such as the output error (OE) method [i.e., the prediction error (PE) method with an output-error model structure] [8], the closed-loop instrumental variable (CLIV) methods [9], the bias-eliminated least-squares (BELS) methods [14]–[16], etc. In particular, the BELS methods have demonstrated their superiority over the OE method and the CLIV method in some important aspects. For example, as a linear regression based algorithm, implementation of the BELS methods is much simpler and less numerically costly than the OE method. The BELS methods can effectively identify closed-loop transfer functions that are probably over-parametrized. In contrast, the OE method exhibits a high sensitivity to

Manuscript received April 6, 2001; revised October 5, 2001 and November 22, 2001. This work was supported in part by the Australian Research Council and in part by the University of Western Sydney, Australia. This paper was recommended by Associate Editor X. Yu. The author is with the School of Quantitative Methods and Mathematical Sciences, University of Western Sydney, Penrith South DC, NSW 1797, Australia (e-mail: [email protected]). Publisher Item Identifier S 1057-7122(02)03130-6.

common factors. Because of the way of constructing the instruments used, the CLIV methods likely suffer serious performance degradation when the system is corrupted by high noise. Thus, they are not so robust against noise as the BELS methods. In [13], some work has been achieved to analyze the connection between an IV type of estimation and the BELS methods in the context of indirect closed-loop identification. Advances have also been made recently in other areas of closed-loop identification. For instance, a strong connection between the indirect closed-loop identification approach and the dual-Youla parametrization based approach is established in [10] in the context of identification of multivariable systems. A unified framework for recursive plant model identification in closed loop is provided in [4], together with several useful validation tests for the models identified in closed loop. New variance results are derived in [7] for various closed-loop identification methods, which are useful not only for theoretical study but also for practical applications. Persistent closed-loop identification of time-varying systems is investigated in [12], with an emphasis on achieving persistently small identification errors with respect to all time. The focus of this paper is on closed-loop identification of an open-loop (possibly unstable) plant which is stabilized by a low-order regulator. As illustrated in [15], the motivation for this problem is that in real-world situations, use of low-order regulators can considerably reduce complexity of the control systems design [1]. The BELS method presented in [14], which can not treat the case of low-order regulators, is modified in [15] and [16]. The main modification in [15] is that a low-pass prefilter of a proper order is designed for prefiltering the measured data, and the known regulator as well as the designed prefilter are combined together to provide necessary information for implementing the bias correction procedure. In this way, unbiased parameter estimates can still be obtained within the BELS framework in the case of inaccurate noise model. Since for the BELS method with prefiltering presented in [15] (termed the BELSP method) extra computations are brought about due to prefiltering of the measured data, it is proposed in [16] that the computational load can be decreased by using the BELS method with no prefiltering (termed the BELSNP method). However, a close examination of the BELSNP method reveals that it still requires treating a high-order augmented closed-loop system. This is rather undesirable because, instead of handling the underlying closed-loop system only, identifying a high-order augmented closed-loop system surely causes an unwanted increase in the computational complexity of the algorithm. In this paper, the BELSNP method is further modified with a view to improving algorithm computational efficiency and achieving a direct closed-loop system estimate. The development of a BELS based method with modified structure is like this. In the first step, the conventional LS method is applied to find an estimate of the underlying closed-loop system, the noise-induced bias of which is shown to be asymptotically determined by the covariance vector between the system output and the disturbance. In the second step, following the same derivation as given in [14], the known regulator is used to form a certain number of linear equations with regard to that noise covariance vector. In the third step, which is very crucial, the additional equations required for solving for that noise covariance vector are derived by appropriately casting an LS estimate of an intermediate parameter vector purposely introduced. In the fourth step, two sets of linear equations obtained respectively in the second and third steps are combined together to arrive at an unbiased estimate of that noise covariance vector. Finally, in the fifth step, the well-known bias correction principle is applied to achieve a direct

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 4, APRIL 2002

557

yet unbiased estimate of the underlying closed-loop estimate, from which an unbiased open-loop plant estimate is then computed using the knowledge of the regulator. Since all redundant computations caused by actual identification of a high-order augmented closed-loop system are now taken away, the modified algorithm is computationally efficient, which is also corroborated with simulation results. II. BACKGROUND As a general setup, we consider the feedback system depicted in Fig. 1. The true plant is described by

y (t) = G(q01 )u(t) + e(t)

(1)

Fig. 1.

Closed-loop system configuration.

regulator C (q 01 ). Define the open-loop parameter vector regulator parameter vectors and , respectively, as

p q  > = [a> ; b> ] = [a 1 1 1 an ; b 1 1 1 bn ] p> = [p 1 1 1 pm]; q> = [q q 1 1 1 qm]:

where y (t) stands for the plant output, u(t) the plant input, and e(t) the stochastic disturbance. The open-loop transfer function G(q 01 ) is given by

01 01 0n G(q01 ) = B (q01 ) = b1 q 0+1 1 1 1 + bn q 0n (2) A(q ) 1 0 a1 q 0 1 1 1 0 an q where q 01 is the unit backward shift operator. The plant is further as-

1

(3)

y(t) = G (q01 )v(t) + (t) where the closed-loop transfer function G (q 01 ) is given by

2

2 = M 0 

1

1

1

1

1

1

( +

+

1

1

+

1

)

( +

)

(6)

and the output error  (t) is defined by

01 01 (t) = A(q )P01(q ) e(t): (7) A(q ) It is assumed that the regulator C (q 01 ) stabilizes the open-loop plant [i.e., A(q 01 ) is a Hurwitz polynomial], and C (q 01 ) itself is known.

M = P0c 0QPcc 2 n > = [p> ; 0] 2 n m

Indirect closed-loop identification consists of two steps. The first step is concerned with identifying the closed-loop system (5) using measurements of the reference signal v (t) and the output y (t). Let us define the closed-loop parameter vector as

2> = [>; > ] = [ 1 1 1 n m; 1 1 1 n m] (8) ^ an estimate of 2 obtained with some identification and denote by 2 1

+

algorithm. The second step of indirect identification is to apply the estimate ^ to solve for the open-loop parameters using the information of the

2

22n (12)

Pc and Qc are (n + m) 2 n Sylvester matrices expanded by p>]> and q> , respectively. If P (q0 ) and Q(q0 ) are coprime, then M is guaranteed to be of full column rank, namely, rank[M] = and

1

[1

2n.

1

Thus, an estimate of  may be determined in view of (11) as ^ +  ): ^ = (M> M)01 M> (2

> = [y > ; t

t

(13)

vt> ] = [y(t 0 1) 1 1 1 y(t 0 n 0 m);

v(t 0 1) 1 1 1 v(t 0 n 0 m)]:

(14)

We may rewrite the output error model (5) for the closed-loop system in linear regression form

y(t) = >t 2 +  (t)

(15)

where the equation error  (t) is given by

 (t) = A(q01 )P (q01 )e(t):

2 2 2^ LS = R0 R y > ] and R y = E [ y(t)]. t t

(16)

As presented in [2], the LS estimate of may be calculated as the minimizing argument of the LS criterion J ( ) = E [ (t)2 ]. This results in

2 +

(11)

2 +2

III. INDIRECT CLOSED-LOOP IDENTIFICATION

1

(10)

Let us consider applying the conventional LS method in indirect closed-loop identification (actually in its first step). Define the regression vector t as

1

1

where

(5)

0 0 0 G (q0 ) = AB((qq0 )) = A(q0 )PB(q(0q ) +)PB(q(q0) )Q(q0 ) q0 + 1 1 1 + n m q0 n m = 1 0  q 0 0 1 1 1 0 n m q 0 n m

(9)

In the light of the structure of the closed-loop transfer function G (q 01 ) given in (6), it is straightforward to show that the closed-loop parameter vector is related to the open-loop parameter vector  via the equation

(4)

and v (t) denotes the reference signal which is assumed independent of the noise e(t). It follows from Fig. 1 that the closed-loop system can be represented by

1

0 1

(2 +2m)

01 01 0m C (q01 ) = Q(q01 ) = q0 + q1 q01 + 1 1 1 + qm q0m P (q ) 1 + p1 q + 1 1 1 + pm q

1

1

1

sumed to be controlled by a regulator

u(t) = v(t) 0 C (q01 )y(t) where the regulator transfer function C (q 01 ) is given by

 and the

1

R

(17)

where = E[ t According to the assumptions imposed on the closed-loop system under study, we may establish the following asymptotic result (see [2] for detail)

2^ LS = 2 + R0 DRy 1

(18)

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where

R

 =

E [ t  (t)] =

=

v

Ry 0

H

E [yt  (t)] E [vt  (t)]

In m Ry = DRy: 0 +

=

0

H>2 = 0H> H>2^ LS = H> 2 + H>R 0 DRy :

(19)

H>R 0 D R^ y = H> (2^ LS + ): 1

R

R

R

(20)

R

This analysis demonstrates the failure of applying the conventional LS method in indirect closed-loop identification in the presence of colored noise. IV. BELS ALGORITHM WITH MODIFIED STRUCTURE In this section, we present a BELS based algorithm that can achieve an unbiased parameter estimate of in the very first step of indirect closed-loop identification. If we assume that the noise covariance vector y is known or an estimate of it is already available, then applying the well-known bias correction principle to (18) gives rise to the following BELS estimate of :

2

1

> = n+m+1 1 1 1 2n

It is seen from (21) that the core of the BELS scheme is estimation of

quently an unbiased estimate of  . In the light of [14, Proposition 1], we may explore the full column and construct a (2n + rank property of the (2n + 2m) 2 2n matrix such that 2m) 2 2m matrix

M

H>M = 0 H

where rank[ ] = 2m. Specifically, lowing way as:

(22)

2^ LS = R 0 R

H may be constructed in the fol-

1

R

1

1

1

1

+

(28)

y

>

=E

t t

R

y =

E

(29)

> ; v> ; t t > vt = [v(t 0 n 0 m 0 1) 1 1 1 v(t 0 2n)]: t

1 1 1 + n mq0 n m + n m q0 n m + 1 1 1 + nq0 n 1 0  q 0 0 1 1 1 0 n m q 0 n m : +

yt

t ( )

>=

1

1

(27)

where

2

0 0 G (q0 ) = AB((qq0 )) = q

1 1 1 0] 2 n0m :

2

H = WU I 0m (23) where W = I n m 0 M(M> M)0 M> , and U is a (2n + 2m) 2 (2n + 2m) nonsingular transformation matrix such that the first 2m columns of WU are linearly independent. Note that the matrix H can 2 +2

= [0

Note that for convenience sake, the variables newly introduced in the auxiliary closed-loop model are labeled by an overbar. As shown in (27), a zero value has been assigned to each of the introduced n 0 m parameters, n+m+1 ; . . . ; 2n , in the numerator polynomial B(q 01 ). This obviously ensures that the two closed-loop systems described respectively by G (q 01 ) and by G (q 01 ) exhibit the same input–output behavior. On the other hand, they are considerably different in that the numerator polynomial of G (q 01 ) is of higher degree 2n while the numerator polynomial of the latter is of original degree n + m. Like (17), the conventional LS estimate of the parameter vector of the auxiliary closed-loop transfer function G (q 01 ) is given by

(21)

Ry , which is vital to obtaining an unbiased estimate of 2 and subseH

R

2> = 2>; >

R

BELS

R

R

2

2 ^ LS 0 R0 DRy : =2

(25)

The above expression represents a system of 2m linear equations with respect to the (n + m)-dimensional noise covariance vector y . It is assumed in [14] that m  n, so y may be unbiasedly estimated by simply using the knowledge of the regulator C (q 01 ) to solve (25). The common and practical situation is that the plant is controlled by a low-order regulator (i.e., m < n) (see [1]). In such cases, (25) alone can not supply enough information for determining y , and thereby additional (n + m) 0 2m = n 0 m equations are needed. In [15], a digital prefilter of order n 0 m is introduced to provide extra information, combined with the knowledge of C (q 01 ), for deriving an unbiased estimate of y . Although no use of a prefilter is made in [16], actual identification of a high-order augmented closed-loop system is yet needed there. In the following, we still treat the case of m < n, but propose a more efficient approach for estimation of the noise covariance vector y . In order to arrive at the desired additional equations for finding y , let us study how the closed-loop system (5) can be identified but with a different selection of the numerical polynomial degree. For this purpose, an auxiliary transfer function is introduced in (26) shown at the bottom of the page where

2

^ LS +  ) ^LS = (M> M)01 M> (2 > 0 1 > 01 = (M M) M [(2 + R DRy ) +  ] 01 > 0 1 > = (M M) M [(M 0  ) + R DRy +  ] > 01 > 01 =  + (M M) M R DRy :

(24b)

Substituting (24a) into (24b), we obtain

2

R DR

(24a)

1

Note that in (19) E [ t  (t)] = is due to the independence between v(t) and e(t) [see also (16)]. It is clear from (18) that the colored noise  (t) [or e(t)] induces a bias 01 y in the LS estimate ^ LS , and this asymptotic bias is decided by the (n + m)-dimensional noise covariance vector y . As shown above, in the second step of indirect identification, determination of an estimate of the open-loop parameter vector  from ^ in terms of (13) is purely an algebraic manipulation. Hence, it is certain that the use of ^ LS in (13) always yields a biased estimate of  . For example, substitution of (18) and then (11) into (13) gives

2^

M

orthogonal also be computed more directly by simply choosing the matrix. Premultiplying (11) and (18) with > , respectively, we have

( +

1

)

1

+

+1

+

( +

+1)

( +

)

2

(30)

2

(26)

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It follows from an analysis similar to the derivation of (18) that

2^ = 2 + R 01 DR (31) > 2 where D = [I 0] 2 . A comparison between ^ (31) and (18) immediately reveals that the bias in 2 and the bias in 2^ are determined by the same noise covariance vector R . LS

Step 3) Apply (17) to calculate the closed-loop parameter LS estimate

y

(n+m)

n+m

LS (

y

R R R = RR> RR R = RR (32) > > where R = E [ v ], R = E [v v ] and R = E [v y (t)]. Furthermore, the matrix inversion formula is used to derive the inverse like of R R0 R0 + R0 R 10 R> R0 0R0 R 10 = 010 R> R0 10 y

v

vv

v

v

y

t t

vv

t t

vy

1

1

1

1

v

1

1

v

1

v

1

v

1

1

(33)

1 R

R R R

>v 01 where = vv 0 (31), respectively, gives rise to

v.

Substituting (33) into (28) and

^ LS = 0101 R>v R01 R y + 101 Rvy ^ = 0101 R> R01 DRy:

(34b)

It then follows from (34a) and (34b) that

R> R0 D R^ 1

v

y =

R> 2^ 0 R v

LS

vy

:

R

R H> R0 D R^ = H>(2^ + ) R> R0 D R> 2^ 0 R the solution of which will give an unbiased estimate of R 1

v

LS

y

1

v

LS

(36)

vy

y . For batch indirect identification of closed-loop systems with low-order regulators, the BELS algorithm with modified structure (termed the BELSM method) can now be summarized as follows. The BELSM Algorithm (Off-Line Version): Step 1) Apply the known regulator to compute the matrix in terms of (23). Step 2) Apply a finite number of input–output measurements fv(t); y(t); t = 1; . . . ; N g to evaluate covariance estimates

H

R^ R^

(N ) =

N) =

v(

1

N 1

N

N t=1 N t=1

>;

t t

R^

v>; R^

t t

N) =

y(

N) =

vy (

1

N 1

N

N t=1 N

t y (t)

1

v

N ) + ) : ^ vy (N ) LS (N ) 0 R

v

(40)

Step 5) Apply (21) to calculate the closed-loop parameter BELS estimate:

2^

^ LS (N ) 0 R ^ 01 (N )DR ^ y (N ): N) = 2

BELS (

(41)

Step 6) Apply (13) to compute the open-loop parameter BELS estimate

^BELS (N ) =

M>M 0 M> 2^ 1

N) +  :

BELS (

(42)

The proposed BELSM algorithm may be implemented recursively in on-line closed-loop identification, since the sample covariance estimates given in (37) and (38) all can be calculated in a recursive manner. V. ANALYSIS AND COMPARISONS First of all, we state the following consistent result, the proof of which is, in some way, different from that presented in [14]. Theorem 1: Under the assumptions imposed on the closed-loop system (5), the BELSM algorithm guarantees that

!1 2BELS (N ) = 2 ^BELS (N ) =  lim  N !1

lim N

^

w.p. 1

(43)

w.p. 1:

(44)

Proof: Premultiplying (15) with v (t 0 k) and taking the mathematical expectation gives

rvy (0k) = 1 rvy (1 0 k) + 1 1 1 + n+m rvy (n + m 0 k) + 1 rvv (1 0 k ) + 1 1 1 + n+m rvv (n + m 0 k ) where the covariance functions spectively, by

(45)

rvy (0k) and rvv (k) are defined, re-

rvy (0k) = E [v(t 0 k)y(t)] rvv (k) = E [v(t 0 k)v(t)]:

(46)

Note that the independence between v (t) and  (t) is used in derivation of (45). Letting k = n+m+1; . . . ; 2n in (45) and writing the resultant n 0 m equations in matrix–vector form yields

R

vy =

(37)

R> 2:

(47)

v

It follows from (39), (17), and (18) that

v y(t): t

t=1

1

1

LS (

(35)

Note that (35) contributes another n–m linear equations in regard to the noise covariance vector y . Hence, by combining (25) and (35) together, we have derived the following system of n + m linear equations with respect to the (n + m)-dimensional noise covariance vector y

(39)

H>R^ 0 (N )D 0 R^ > (N )R^ 0 (N )D H> (2^ 1 ^> ^ R (N )2

N) =

y (

(34a)

v

LS

R^

t

vy

N ):

y(

Step 4) Apply (36) to compute the noise covariance estimate:

LS

We now show that the desired additional equations may be obtained by the way of finding from (28) and (31) two expressions for the LS estimate of the introduced parameter vector . The special form of the regression vector t defined in (30) may be first employed to partition the covariance matrix and the covariance vector y as

^ 01 (N )R ^ N) = R

2^

(3n+m)

LS

559

(38)

01 ^ !1 2LS (N ) = R R

lim N

2

y = ^ LS =

2 + R0 DR 1

y

:

(48)

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Letting N

!1 Ry (N ) H>R01 D 01 = R> R01 D

lim

N

! 1 in (40) and using (48), (24a), and (47) leads to ^

v

=

H>R01 D R> R01 D H>R01 D R> R01 D v

=

For this simulation, the open-loop transfer function is given by

v

01 01

G(q

H> (2^ LS + ) R>v 2^ LS 0 Rvy

H>(2 + R01 DR + ) R> (2 + R01 DR ) 0 R > 01 (H R D)R =R : > 01 (R R D)R

01 ) =

y

y

v

y

C(q

vy

y

(50)

whose two poles are located at z = 2:4173 and z = 0:0827. A firstorder regulator represented by

y

v

01 0 0:6q02 1 0 2:5q 01 + 0:2q 02 q

(49)

Finally, (43) and (44) follow straightforwardly from (41), (42), (48), and (49) Next, we make comparisons between the BELSNP method presented in [16] and the proposed BELSM method. One substantial algorithmic difference between them is that the BELSNP method involves identifying the high-order augmented closed-loop system with its numerator polynomial degree being increased from n + m to 2n, where n > m. That is, it has to estimate an augmented (3n + m)-dimensional parameter vector of the high-order augmented closed-loop system in its identification procedure. On the contrary, the BELSM method essentially works directly with the underlying closed-loop system (5) with the original model orders. This is because in the derivation of the BELSM method, the auxiliary closed-loop model with G (q 01 ) is introduced only as the mechanism to attain an estimate of the noise covariance vector y , but there is no necessity to really identify it at all. This leads to the elimination of all the redundant calculations incurred by the BELSNP method. Let L denote the total number of estimated parameters in the system model. The computational complexity of a parametric algorithm may be characterized primarily by the updating of a matrix, and it will grow with L2 . Considering dimensionality of updated matrices used, the computational load of the BELSM method and the BELSNP method is roughly proportional to (2n+2m)2 +(n+m)2 and to (3n + m)2 + (n + m)2 , respectively. As a typical example, let us assume that n = 2m, that is, the stabilizing regulator is of an order equal to half that of the open-loop plant. Then the BELSM method can achieve an about 22% reduction in the computational load in comparison with that of the BELSNP method, which is quite significant. The other subtle algorithmic difference is that the BELSM method yields an unbiased estimate of the original parameter vector in a very direct manner, while the BELSNP method takes the first 2n + 2m elements of the augmented closed-loop parameter vector as the BELS estimate of . Finally, we remark that like the BELS based algorithms presented in [14]–[16], the BELSM method can produce unbiased parameter estimates with no need to model the noise contribution on the measured data, thereby having a strong robustness against noise. Moreover, it is superior to the OE method in two aspects: considerably less computationally intensive and insensitive to common factors that likely exist in closed-loop transfer functions. On the other hand, the BELSM algorithm has a significant advantage over the CLIV method due to its good reliability of working under very low values of signal-to-noise ratio (SNR), though the former requires a few more computations than the latter.

R

2

2

VI. A SIMULATION EXAMPLE A simulated example is used in this section to verify the performance of the proposed BELSM algorithm. In particular, an open-loop unstable plant is studied to show that the stability of the open-loop plant is not a necessary condition for application of the algorithm.

01 ) =

01 1 0 0:6q 01

1 + 0:5q

(51)

is used to stabilize the open-loop unstable plant. In view of (6), the closed-loop transfer function is found to be

01 02 + 0:36q03 G (q01) = 1 0q2:1q0011:2q : + 1:6q 02 0 0:42q 03

(52)

It is noticed that G (q 01 ) contains a common factor (1 0 0:6q 01 ). The reference input v(t) is generated as the following moving average process v(t) = (1

0 0:2q01 + 0:4q02 0 0:6q03 + 0:8q04 ) !(t)

(53)

where !(t) is zero-mean white noise with unit variance. The colored noise e(t) acting on the system is described by e(t) =

0 1:8q01 + 1:5q02 0 0:66q03 + 0:1q04 (t) 1 0 1:7q 01 + 0:24q 02 + 1:072q 03 0 0:576q 04 1

(54)

where (t) is a white noise sequence with zero mean and variance 2 , and is independent of !(t). In view of (5), the SNR at the output of the closed-loop system is defined as SNR = 10 log10

G 01 )v(t))2

E ( (q

E[(t)2 ]

(dB)

(55)

and the value of SNR is varied by changing the noise variance 2 . For comparison, application of the following eight algorithms in batch identification of the above closed-loop system is considered. The conventional LS method, the OE method, the PE method (i.e., the PE method applied to the closed-loop system with a general linear model structure, and it is usually called PEM), the basic IV (BIV) method that adopts the lagged reference signal v(t) as instruments, the IV4 method that represents the optimal four-stage IV method [8], the BELSP method that employs a prefilter designed as F (q 01 ) = 1=(1 0 0:75q 01 ), the BELSNP method, and the proposed BELSM method. All the eight methods are implemented via the two-step indirect closed-loop identification procedure, with (13) being used to access the open-loop parameters. The simulations are conducted in the MATLAB environment, with the OE, PEM and IV4 methods being implemented via the MATLAB codes oe, pem and iv4, respectively [6]. Note that the structure of the disturbance e(t) as given in (54) is incorporated into the closed-loop model structure when PEM is applied. The performance of these eight identification algorithms is compared in terms of the relative error (RE), the normalized root mean squared error (RMSE), and the number of flops. For example, for the closed-loop system, the RE and RMSE are defined, respectively, as

RE =

km(2^ ) 0 2k k2k

m(2^ ) = K1

K

2^

k

k=1

RMSE =

1 K

K k=1

k2^ k 0 2k2 ; k2k2 (56)

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561

TABLE I RELATIVE ERRORS (RE) FOR CLOSED-LOOP SYSTEM (

N = 500, 500 MONTE-CARLO TESTS)

TABLE II ROOT MEAN SQUARED ERRORS (RMSE) FOR CLOSED-LOOP SYSTEM (

N = 500, 500 MONTE-CARLO TESTS)

TABLE III RELATIVE ERRORS (RE) FOR OPEN-LOOP PLANT (

N = 500, 500 MONTE-CARLO TESTS)

TABLE IV ROOT MEAN SQUARED ERRORS (RMSE) FOR OPEN-LOOP PLANT (

N = 500, 500 MONTE-CARLO TESTS)

TABLE V

N = 500, 500 MONTE-CARLO TESTS, SNR = 5 dB)

COMPUTATIONAL LOAD (

2

where ^ k denotes the parameter estimator in the k th Monte-Carlo test over a total of K tests. The MATLAB code flops, which counts the number of floating point operations, is used to describe the computational efficiency of an estimation algorithm. The length of the measured data is N = 500 data points, and K = 500 Monte-Carlo tests are performed. In order to examine the robustness of the algorithms against noise, the variance 2 is selected such that the SNR takes the value of 20 dB, 10 dB, 5 dB, 1 dB, 0 dB, 01 dB, 0 5 dB, and 010 dB, respectively. The RE and RMSE with regard to

the estimates of the closed-loop parameters are listed in Tables I and II, whereas the corresponding RE and RMSE for the open-loop parameter estimates are shown in Tables III and IV. As an illustration, Table V gives the numerical cost in the case of SNR = 5 dB. In accordance with our theoretical assertions, the conventional LS method is seriously biased with high variance due to the colored noise. This example, together with the other two different examples presented in [14]–[16], further confirms our observation that as a prediction error based method, a strong sensitivity to common factors is inherent in the

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OE method. Needless to say, its implementation also demands a very large amount of computations. It is seen that PEM fails to yield unbiased and statistically efficient estimates of the closed-loop general linear model for this simulated example even though it takes into account the structure of e(t). Moreover, since it involves modeling of the process noise, PEM is highly computationally demanding. Both the BIV method and the IV4 method give entirely useless estimates. The reason behind this is that they are also sensitive to common factors so as not to be able to deal with over-parametrized models. These observations illustrate that it is not possible to identify this closed-loop system with the transfer function G (q 01 ) given in (52) by any standard method because of the fact that the pole at z = 0:6 is not observable from the system output. On the contrary, one can clearly observe that the three BELS based methods outperform the other five algorithms in terms of consistency and low variance, robustness against noise, and insensitivity to common factors. While the performance of the BELSM method is comparable to that of the BELSP method and the BELSNP method, its computational advantage is also obvious. Due to no use of prefiltering of the measured data and no handling of a high-order augmented closed-loop system, the BELSM method has decreased the computational load by approximately 25% from that of the BELSP method. In contrast, the BELSNP method has decreased the numerical cost by only around 8% from that of the BELSP method. All these justify the theoretical prediction made in the preceding section on the computational efficiency of the proposed BELSM method.

VII. CONCLUSIONS In this brief, a modified version of the BELS method has been proposed that can be used for indirect identification of transfer function models for unstable plants in closed loop, given the knowledge of loworder regulators. The main feature of the proposed BELSM method is that the underlying closed-loop system is identified in a direct manner, with no need to deal with a high-order augmented closed-loop system, thus offering some important algorithmic advantages over the previous BELSNP method. In particular, the computational cost can be significantly reduced with the BELSM method. The simulated experiments have shown that the proposed algorithm can produce parameter estimates with a very good accuracy in highly noisy environments. It is reasonable to expect that good plant estimates given by the BELSM

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