Parametric identification of linear systems operating ... - IEEE Xplore

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 48, NO. 4, APRIL 2001

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Parametric Identification of Linear Systems Operating Under Feedback Control Wei Xing Zheng, Senior Member, IEEE

Abstract—This paper considers the indirect identification of closed-loop systems with colored noise when there exist low-order controllers in the loop. A new form of bias-eliminated least-squares (BELS) method is developed which is based on a simple and efficient technique of estimating the colored-noise-induced bias through a combined use of the known information about the controller and a proper number of zero parameters introduced. The main feature of the presented BELS based method is that no prefiltering of sampled data is needed and an unbiased estimate for the closed-loop parameters is given in a direct manner. The presented method is verified via application to indirect identification of an open-loop unstable plant. Index Terms—Closed-loop identification, feedback systems, least-squares method, parameter estimation.

I. INTRODUCTION

I

DENTIFICATION of a plant in closed-loop operation is an issue of practical significance. There are two important aspects of closed-loop system identification. These are : 1) that it is mandatory for unstable plants and 2) that the closed-loop setting functions somewhat like an attenuating mechanism which counters the disturbance effect. One of the commonly used closed-loop identification approaches is the method of indirect identification [5], [10]. It is based on the idea of first identifying a closed-loop transfer function and then calculating the related plant model by use of the knowledge of the present controller in the loop. In the case of the inaccurate noise model, the parameter estimates given by the conventional least-squares (LS) method are bound to be biased. Recently, the bias-eliminated least-squares (BELS) method is proposed to perform unbiased identification of closed-loop systems via the indirect procedure [14]. The BELS method is built on the bias correction principle [3] by a novel and efficient use of the given knowledge of the controller. One feature with the method is that there is no need to model the noise contribution on sampled data. The computational load with the BELS method is less than that of the output error (OE) method, (i.e., the prediction error (PE) method with an output-error model structure) [10], as the OE method involves iterative nonlinear optimization procedures. Moreover, the BELS method possesses better robustness against noise than Manuscript received April 26, 2000; revised October 30, 2000. This work was supported in part by a Research Grant from the Australian Research Council, and in part by a Research Grant from the University of Western Sydney, Nepean, Australia. This paper was recommended by Associate Editor G. Chen. The author is with the School of Science, University of Western Sydney, Nepean, Kingswood, Sydney, NSW 2747, Australia (e-mail: [email protected]). Publisher Item Identifier S 1057-7122(01)02873-2.

the closed-loop instrumental variable (CLIV) methods [11], since it is observed that the CLIV methods fail to work efficiently when the noise level is high. However, when open-loop plants are being controlled by low-order controllers, which is very common in practical circumstances due to the considerations of low complexity [1], the BELS method presented in [14] is not applicable. It is proposed in [15] that in such cases a low-pass prefilter with a proper order can be designed to process the sampled data. Then, through a combined use of the known information of the controller and the designed prefilter, the BELS method can still be implemented to obtain unbiased parameter estimates, irrespective of noise dynamics. In the sequel, the BELS method with prefiltering presented in [15] is called the BELSP method for convenience of illustration. Aside from the OE method, the CLIV method and the BELS method mentioned above, the other recent developments in closed-loop identification can be found, for example, in [9], [4], [12], [7], and [13], to just mention a few. A frequency-domain identification method is proposed in [9] which takes into account the correlation between the input and output disturbances. On the basis of the output error model structure, a unified presentation of recursive algorithms for plant model identification in closed loop is given in [4]. For identification of multivariable systems it is shown in [12] that the indirect method of closed-loop identification and the dual-Youla parameterization based approach are closely related to each other. 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 [13], with an emphasis on achieving persistently small identification errors with respect to all time. The purpose of this paper to consider unbiased identification of a linear plant operating in closed loop with a low-order digital controller in the presence of a misspecified noise model. The BELSP method presented in [15] is certainly applicable to this problem for indirect closed-loop identification. However, two issues exist with this method which are worth further studying. One issue is that design of a prefilter is required and prefiltering of sampled data is then performed, which causes additional computations. The other is that the parameter estimates of the underlying closed-loop system have to be recovered from those of one augmented system, which is not convenient, for example, when one needs to use the closed-loop poles for model validation in terms of pole closeness. To overcome these drawbacks, a new form of BELS method with no prefiltering (denoted by BELSNP for short) is developed in this paper which is based on important modifications made

1057–7122/01$10.00 © 2001 IEEE

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on the BELSP method presented in [15]. The basic idea of the BELSNP method is to increase the denominator polynomial of the transfer function of the underlying closed-loop system by the dimension that is equal to the difference of the order of the open-loop plant and the order of the controller. Then, the resulting augmented closed-loop system contains some known parameters that take zero as its true values. The parameters of the augmented closed-loop system are estimated by applying the conventional LS method directly to the sampled data. Using straightforward asymptotic analysis, a simple relationship between the true and the LS augmented closed-loop parameters is derived in terms of the given controller parameters and the introduced known zero parameters, which lays down a basis for estimating the colored-noise-induced bias. Finally, unbiased estimates of the closed-loop parameters as well as the open-loop parameters are obtained by means of the bias correction principle. A simulation example is given to illustrate the performances of the proposed BELSNP method.

Fig. 1.

Closed-loop system configuration.

Fig. 1 gives the closed-loop system configuration, where denotes the plant output, the plant input, the external the stochastic disturbance. The aim is to identify signal, and the transfer function of the open-loop plant described by

In addition to the assumption of the known controller , is observable and persistently exwe further assume that 1) and are mutually uncorciting of a sufficient order, 2) is a Hurwitz polynomial, 4) there related statistically, 3) and , and between exist no common factors between and , and 5) the orders and are known. Besides, it is noticed from (1) and (2) that there is an implicit assumption that the plant and controller models have each equal numerator and for and denominator polynomial orders [i.e., for ]. However, this assumption is introduced purely for notational convenience, and can be easily relaxed to include the case where the orders of numerator and denominator polynomials are different. In indirect closed-loop identification, is first estimated by the conventional LS method

(1)

(9)

II. CLOSED-LOOP IDENTIFICATION PROBLEM

where under the assumption of the known controller which is defined by

in the loop, (10) (2)

With this system configuration, the transfer function the closed-loop system can be shown to be

of

(11) (3)

where

and

Note, that for convenience of illustration, the asymptotic case of actually reprethe infinite sample size is treated here, and sents the expected value of the LS estimate. Under the imposed assumptions it can be shown (see e.g., [2]) that

where (12)

are obtained from the following equation:

(13)

(4) The input–output behavior of the closed-loop system may then be represented by the following linear regression: (5) , the closed-loop parameter vector where the equation error , and the regression vector are defined, respectively, as (6) (7)

(8)

is biased when is colored noise, there Since by (11) is no doubt that the second step of the indirect identification procedure will give rise to a biased estimate of the open-loop parameter vector (14) when is calculated from parameters

by using the known controller (15)

Note, that in order to minimize the transformation effect at the second step of the indirect identification procedure, the open-loop parameters are usually determined from the

ZHENG: PARAMETRIC IDENTIFICATION OF LINEAR SYSTEMS OPERATING UNDER FEEDBACK CONTROL

estimated closed-loop parameters via the standard exact transformation defined as

453

Similarly, to (9), the conventional LS estimate of the augmented closed-loop parameter vector is given by (20)

(16)

where (21)

By means of the bias correction principle [3], the BELS estimator of can be obtained as follows:

(22)

(17) is known or an provided that the noise covariance vector estimate of it is already available. It turns out that an unbiased estimate of , and consequently an unbiased estimate of , will can be estimated in certain manner. This is the result if , it is shown in [14] kernel of the BELS method. When provides sufficient information that the known controller . But when , that is, when the system for estimating can be is with a low-order controller, it is shown in [15] that and simultaneously estimated by using the knowledge of to prefilter the applying a designed digital filter of order . In the sequel, a more straightforward input measurements . technique is presented for estimation of

Further, similarly to (11), we can have (23) . We note that the where colored-noise-induced bias in (23) is determined by the same as in (11). noise covariance vector conSince the augmented closed-loop transfer function , from (2), (3), (14), (15), (18), tains the known controller and (19) we can establish the following relationship between the augmented closed-loop parameter vector and the open-loop parameter vector

III. INDIRECT IDENTIFICATION WITH NO PREFILTERING First of all, we augment the closed-loop numerator polynoby a dimension of , with the introduced parammial eters all taking zero as its true values. This yields the augmented closed-loop transfer function defined by (18) at the bottom of the page where

(24) where (25) (26)

(19) For convenience of illustration, the corresponding variables in the augmented system are marked with an overbar. It is important to note that although the augmented is equivalent to the closed-loop transfer function in the sense that underlying transfer function , the essential difference between them is that the numerator polynomial of the former is of while the latter of order . It is these introduced order in , plus the known zero parameters , that enables us known information about the controller in the presence of to estimate the noise covariance vector a low-order controller in the loop.

(27)

.. . .. .

.. .

.. .

..

.

..

.

..

.

..

.

..

.

.. .

(28)

.. .

(18)

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We can then implement unbiased estimation of .. . .. .

.. .

.. .

..

.

..

.

..

.

..

.

..

.

.. .

by means of (39)

(29)

.. .

and guaranNote, that the coprimeness between as well as are of full column rank, namely, tees that rank (see e.g., [10]). rank fullNow, we are going to construct a column-rank matrix which meets the condition (30) To this end, we let

In practical circumstances, only a finite number of input–output measurements can be collected. So, the unknown covariances and in the theoretical equations (20) and (37) need to be replaced by its natural estimates

(40) Hence, we can now establish the following BELSNP method for off-line and on-line indirect identification of closed-loop systems with low-order controllers. Off-Line Algorithm: Step 1) Find the matrix by use of (31)–(33). Step 2) Estimate by applying the conventional LS method and to the sampled data utilizing (20) and (40), which gives

(31) where

(41) Step 3) Calculate the noise covariance vector which gives

via (38),

(32) is a and matrix such that the first independent. This leads to matrix as

nonsingular transformation columns of are linearly . We define the

where is an arbitrary forward to show that rank

Step 4) Perform the bias correction via (39), which gives (43)

(33)

rank

(42)

matrix. It is then straightrank (34)

Further, it is easy to verify that

Step 5) Compute

the

open-loop parameter estimate from via the standard exact transformation (16). On-Line Algorithm: Step 1) Compute the matrix by means of (31)–(33). and propStep 2) Set recursive initial values for . erly with Step 3) Use the conventional recursive LS procedure to obtain (see e.g., [8])

(35) so (30) is satisfied. Combining (30) with (24) gives

(44a) (36)

Premultiplying (23) with

and using (36) leads to

(44b) where the gain matrix

is defined as

(37) The above set of solve for the which yields

(44c)

linear equations may be employed to -dimensional noise covariance vector , Step 4) Estimate the noise covariance vector (38)

by use of (45)


Step 5) Carry out the bias correction (46) Step 6) Calculate

from the available via the standard exact transformation

(16). Step 7) If the chosen convergence criterion is satisfied, then and go to step 3. stop; otherwise, set IV. COMMENTS The convergence of the BELSNP method can be analyzed in a similar way to the BELS method presented in [14] and to the BELSP method presented in [15], because the proposed method is still based on the bias correction principle. Moreover, the BELSNP method retains all the merits of the BELSP method as described before. For example, the proposed method requires fewer computations than the OE method whereas it can work more reliably than the CLIV method in the presence of high noise. Further, unlike the OE method, the BELSNP method is not sensitive to common factors that likely exist in closed-loop transfer functions. It is very interesting to make a comparison between the proposed BELSNP method and the BELSP method presented in [15]. Though these two methods both solve a larger identification problem with the augmented closed-loop transfer function of equal orders, one can easily see that in our off-line algorithm are used directly the sampled data when the parameters of the augmented closed-loop transfer are first estimated by the conventional LS function method. Hence, one significant difference between the BELSP method and the BELSNP method is that the BELSNP method works directly with the sampled data. In implementation of the proposed method for unbiased identification, the user is relieved from the task of designing a prefilter and prefiltering the sampled data. It is evident that added computations due to prefiltering can be saved. Besides, the unfiltered data would make more sense for a conventional LS estimate. Furthermore, another attractive feature of the BELSNP elements method is that the BELSNP estimate of the last of given by (39) always takes the zero value, namely (47) consists [see the Appendix for proof of (47)]. Since elements of , the BELSNP of just the first approach is superior to the BELSP method in that it can give a direct unbiased estimate of the closed-loop parameter vector with virtually no need to extract from . This advantage of the BELSNP method can be very useful when the model validation in terms of pole closeness is performed in order to test the quality of the identified closed-loop model [4]. Note that this analysis applies to both the developed off-line

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and on-line algorithms, just with the covariances and replaced by its corresponding estimates. That is, (47) is not merely a theoretical result. Practically, the introduced paramwill also always take the zero values eters while numerical errors have no influence on the performance of the proposed method in this regard. This is because the conelements of the numerical straint (36) can force the last to be zero on all occasions. We finally estimate elements of , remark that the LS estimate of the last , does not necessarily assume the zero value as is i.e., usually biased. V. ILLUSTRATIVE EXAMPLE In this section, the efficiency of the proposed BELSNP method is illustrated by a simulation example. The open-loop plant model considered is

This second-order plant, which is unstable with its two poles and , is stabilized by a first-order located at controller given by

The resultant closed-loop system is described by

which, however, contains a common factor . The system is corrupted by colored noise modeled as shown in the is zero-mean white equation at the bottom of the page where . The external input is set as noise with variance . zero-mean white noise with unit variance, independent of This leads to the signal-to-noise ratio (SNR) at the closed-loop system output terminal to be about 2.36 dB, indicating the presence of high noise. Note, that this example is considered in [15]. Off-line indirect closed-loop identification is performed by using the conventional LS method, the OE method, the CLIV method, the BELSP method, and the BELSNP method. In particular, the CLIV method adopts delayed the set point values as instruments, while for the BELSP method the prefilter is designed as . For comparison purpose, and (see [6]) for the instrumental two MATLAB codes variables (IV) methods are also applied to indirect identification, where the instrumental variables of the IVX method conand the IV4 method represist of the lagged external signal sents the optimal four-stage IV method [10]. The relative error is introduced to give an overall description of estimation accuracy of an estimator . Also, we calculate poles and zeros of the closed-loop system as well as the open-loop

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TABLE I PARAMETER ESTIMATES FOR CLOSED-LOOP SYSTEM (SNR = 2.36 dB,

N = 1000, 500 MONTE-CARLO TESTS)

TABLE II PARAMETER ESTIMATES FOR OPEN-LOOP PLANT VIA THE STANDARD EXACT TRANSFORMATION (SNR = 2.36 dB,

TABLE III PARAMETER ESTIMATES FOR OPEN-LOOP PLANT (SNR = 2.36 dB,

plant by use of the mean values of the parameter estimates. The amount of computations is measured in terms of the MATLAB (count of floating point operations). Five hundred code Monte-Carlo tests are conducted for ensemble average and one thousand data points are used for each test. In Tables I–III, each entry contains three values. The first value is the sample mean of the estimated parameter, the second value the standard error from the sample mean and the third value the standard error from the true value. Note that the use of the standard exact transformation (16) in determination of the open-loop parameters renders the model order to be 4 in Table II. The comparative



performances of these identification methods are displayed in Table IV. As expected, the conventional LS method produces biased parameter estimates, and the two BELS based methods both yield unbiased parameter estimates. The OE method not only is highly computationally demanding, but also fails to work properly because of its sensitivity to the presence of common factors. Surprisingly, neither the IVX method nor the IV4 method is able to give correct parameter estimates for this example. The reason may be also due to its lack of ability of handling over-parameterized models. It is seen that the BELS based methods


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TABLE IV COMPARATIVE PERFORMANCES (SNR = 2.36 dB, = 1000, 500 MONTE-CARLO TESTS)

N

are more accurate than the CLIV method, though the latter requires fewer computations than the former. On the other hand, it is observed that the amount of computations involved with the BELSP method is about 10% more than that of the BELSNP method. The increased computational effort is obviously caused by prefiltering of the sampled data which is required by the BELSP method. Moreover, the BELSNP estimates are desirably accurate though its variances are a bit greater than those of the BELSP estimates.

choice of , or will remain the same for different . Preyields multiplying (39) with (48) Then, substitution of (38) into (48) leads to (49) Using (33) in (49), we obtain (50)

VI. CONCLUSIONS The problem of indirect identification of closed-loop systems with low-order controllers and subject to colored noise has been investigated. The model is obtained by applying a conventional LS estimate of the closed-loop parameters. The parameters are then adjusted with a view to removing the bias caused by colored noise. Such an adjustment is possible by means of the knowledge about the parameters in the controller and the introduced known zero parameters. It has been demonstrated that with no prefiltering of the sampled data, the unbiased parameter estimates can still be obtained by the proposed BELSNP method for the closed-loop system as well as the open-loop plant. In addition to the computational advantage over the BELSP method, another attractive aspect of the BELSNP method is the capability of providing a direct parameter estimate of the underlying closed-loop system, which is useful for the purpose of model validation. APPENDIX A PROOF OF EQUATION (47): Since contains an arbitrary submatrix , (38) actually imcan be estimated in the way independent of the plies that

So (47) is immediately derived from the last of the matrix equation (50).

equations

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Wei Xing Zheng (M’93–SM’98) was born in Nanjing, China. He received the B.Sc. degree in applied mathematics and the M.Sc. and Ph.D. degrees in electrical engineering, in 1982, 1984 and 1989, respectively, all from the Southeast University, Nanjing, China. From 1984 to 1991, he was with the Institute of Automation at the Southeast University, Nanjing, China, first as a Lecturer and later as an Associate Professor. From 1991 to 1994, he was a Research Fellow in the Department of Electrical and Electronic Engineering and the Interdisciplinary Research Centre for Process Systems Engineering at the Imperial College of Science, Technology and Medicine, University of London, London, U.K.; in the Department of Mathematics at the University of Western Australia, Perth, Australia; and in the Australian Telecommunications Research Institute at the Curtin University of Technology, Perth, Australia. He joined the University of Western Sydney, Sydney, Australia, as a Lecturer in 1994, and has been a Senior Lecturer in the same university since 1996. In 1998, he was on sabbatical leave at the Institute for Network Theory and Circuit Design at the Munich University of Technology, Munich, Germany and in the Department of Electrical Engineering at the University of Virginia, Charlottesville, VA. His research interests are in the areas of systems and controls, signal processing, communications, and operations research. He coauthored the book Linear Multivariable Systems: Theory and Design (SEU Press, Nanjing, 1991). Dr. Zheng has received several science prizes, including the Chinese National Natural Science Prize awarded by the Chinese Government in 1991. He is a Member of the IFAC Technical Committee on Modeling, Identification and Signal Processing (1999–2002), and has been an Associate Editor of the Conference Editorial Board of the IEEE Control Systems Society since January 2000.