Frequency Response Estimation for NCFs of an MIMO ... - IEEE Xplore
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 1, JANUARY 2006
Frequency Response Estimation for NCFs of an MIMO Plant From Closed-Loop Time-Domain Experimental Data Tong Zhou, Member, IEEE
Abstract—Frequency response estimation from closed-loop time-domain experimental data is investigated in this paper for normalized coprime factors (NCF) of a multiple-input–multiple-output (MIMO) plant. Based on a linear fractional transformation (LFT) representation for all the NCFs of plants internally stabilizable by a known controller, this estimation problem is converted into the open-loop nonparametric estimation of an inner transfer function matrix (TFM). An estimate is derived through constrained data-matching. It is proved that when the probing signals are periodic and the NCFs of the auxiliary plant are appropriately selected, the estimate is asymptotically unbiased and with a normal/complex normal distribution. It has been made clear that the estimation bias and the estimation variance are always finite. Computationally tractable procedures are suggested for choosing the desirable auxiliary TFMs. Index Terms—Closed-loop identification, multiple-input–multiple-output (MIMO) system, normalized coprime factorization, robust control.
I. INTRODUCTION
I
N INDUSTRIAL applications, it is often required that system identification is performed in closed-loop due to safety and economy considerations, etc. While investigations on closed-loop system identification have a long history, some important issues still need further studies [7], [15], [4], [8]. For example, when an indirect approach is adopted in nonparametric estimation, the estimate is possibly with an infinite variance; when a direct approach is employed in deriving a parametric model, a noise model is required that can describe the actual disturbance characteristics, etc. In representing plant dynamics, many kinds of models have been suggested. Transfer functions and state space models are two instances that are extensively applied in system synthesis. Recently, NCF-based representations became popular, mainly due to its appreciable properties in representing the dynamics of unstable plants, its close relations with the gap metric and the -gap metric [13], [6], [5], [14], [9]. Although model sets with NCF perturbations (NCFPMS) are well utilized in robust control theory, it is not very easy to identify them from experimental data. While great advances have Manuscript received July 21, 2004; revised April 25, 2005 and July 29, 2005. Recommended by Associate Editor E. Bai. This work was supported in part by the National Natural Science Foundation of China under Grant 60174022, and in part by the Trans-Century Training Programme Foundation for the Talents of MOE, P.R.C., Tsinghua University under Grant JC2003058. The author is with the Department of Automation, Tsinghua University, Beijing 100084, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.860279
been made, respectively, in [3], [6], and [12] toward NCFPMS estimation, practical applications of the suggested algorithms can still be hardly regarded as very convenient [16]–[18]. An essential issue here seems to be the lack of mathematically appreciable relations between NCFs and physically measurable variables [9], [6], [3], [12]. To solve this identification problem, it is suggested in [16]–[18] to divide it into two steps. At first, frequency response is estimated for NCFs. Then, a parametric NCFPMS is identified. While theoretical and numerical studies show that this approach is really capable of providing a NCFPMS from experimental data, all the studies are based on frequency-domain experimental data. Moreover, in deriving the analytic expression for the nonparametric estimate, covariance matrices are required for external disturbances and measurement errors. There is, however, no guarantee that this information is always available in practical engineering [8], [10]. In this paper, we discuss time-domain experimental data based nonparametric NCF estimation for MIMO plants in a closed-loop system. For a concise presentation, only normalized right coprime factors (NRCF) are discussed. When plant normalized left coprime factors (NLCFs) are required to be identified, it can be dealt with through coping with the transpose of the plant. Based on an LFT parameterization for all the NRCFs of plants internally stabilizable by a known controller, closed-loop nonparametric NRCF estimation is converted into the open-loop nonparametric estimation for an inner TFM. A procedure is proposed for estimating the frequency response of plant NRCFs, through constrained data-fitting from a batch of closed-loop time-domain experimental data. It has been made clear that the estimation bias and variance are always finite, which is significantly different from nonparametric estimations for the plant itself. Some important statistical properties of the estimate are investigated under the condition that the external disturbances and the measurement errors are zero-mean weak stationary random sequences with finite second order central moments. It is proved that when the NRCFs of the auxiliary plant are appropriately selected and the probing signals are periodic, the estimate is asymptotically unbiased and normally/complex normally distributed. On the other hand, for general probing signals, the asymptotic statistical properties of the estimate are shown to be completely determined by a normalized random matrix with all its columns having normal/complex normal distributions. A computationally tractable method is suggested for choosing the desirable NRCFs of the auxiliary plant, as well as other involved auxiliary
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ZHOU: FREQUENCY RESPONSE ESTIMATION FOR NCFS
Fig. 1.
Closed-loop identification experiment.
TFMs. Some numerical simulation results are reported which confirm the theoretical results. It is worthwhile to point out that in both the frequencyand the time-domain data based closed-loop NRCF nonparametric estimations, the major issues are similar. The estimation algorithms, however, have completely distinctive features. Moreover, there are significant differences in overcoming the involved mathematical problems arising in estimation quality evaluations. Furthermore, some important theoretical conclusions, such as the construction and the selection condition for the desirable auxiliary TFMs, etc., are not completely the same. These differences and difficulties are mainly due to the following facts. When a frequency-domain approach is adopted, the experimental data at different sampled frequencies can generally be assumed independent and normally/complex normally distributed. This property enables us to derive a nonparametric NCF estimate using only the experimental data at the same sampled frequency. However, when a time-domain approach is employed, this assumption is typically no longer valid. The dependence among time-domain experimental data at different sampled time instants makes the analysis of the estimation algorithm much more mathematically involved. On the other hand, although the closed-loop nonparametric NRCF estimation problem has been converted into an open-loop one, the latter is not a trivial application of the existing results. The requirement that the related plant is inner invalidates the available nonparametric plant estimates. The rest of this paper is organized as follows. In the next section, the estimation problem is described and a parameterization is provided for all the NRCFs of plants internally stabilizable by a known controller. The nonparametric estimate is obtained in Section III, as well as its asymptotic statistical properties. The auxiliary TFM selection problem is dealt with in Section IV, while Section V reports some numerical simulation results. The paper is concluded by Section VI. Finally, an Appendix is included which gives proofs of the lemmas and theorems. The notation used in this paper is fairly standard. Its detailed definition is therefore not included, but can be found, for example, in [5], [9], [6], and [16]. II. PROBLEM DESCRIPTION AND PRELIMINARY RESULTS Typically, a closed-loop system used in identification has a structure like that depicted in Fig. 1. Here, and are probing signals designed to obtain some dynamical characterand are, respectively, measureistics of the plant; reprement errors of the plant input and output, while sents the composite influence of exterior disturbances on the closed-loop system. On the other hand, and represent, respectively, the plant and the controller. In this paper, we only
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consider linear time-invariant plants and controllers. It is asinputs sumed that both the plant and the controller are of and outputs, and identification experiments have been performed. Superscript is adopted to clarify that a variable or a function, etc., is related to the th identification experiment, . Moreover, we also assume that the controller is known and of finite McMillan degree. Let and , denote respectively the sampled values of the probing signals and plant input/output measurements in the th identification experiment. The objective is to derive an estimate for the frequency response of the plant NRCFs and analyze its statistical properties. Compared with the frequency-domain experimental data based estimation problem discussed in [16], [17], there are no assumptions in the problem formulation on the statistical properties of the external disturbances and the measurement errors. It will become clear afterwards (Theorems 1 and 3) that a consistent estimate can be derived under some weak conditions. Moreover, the assumptions adopted in the above problem description are quite standard in closed-loop system identification. On the other hand, it is worthwhile to mention that while frequency-domain experimental data are easier to be prefiltered and combined from different identification experiments, time-domain experimental data are generally much easier to be obtained [8], [10]. In this paper, only square plants are dealt with. The results, however, can be extended to nonsquare plants. Moreover, similar results can be derived through completely the same arguments for the nonparametric estimation of plant NLCFs. Similar to the frequency-domain data based closed-loop nonparametric NRCF estimate, a time-domain counterpart is derived in this paper also from a parameterization for all the NRCFs of plants that compose an internally stable closed-loop system with a known controller. To express this parameterizais employed which is internally tion, an auxiliary TFM and has and as its NRCFs.1 stabilizable by Moreover, to make statements concise, the following symbols stands for when are at first defined in which is real and rational. and and Assume that are, respectively, NRCFs and NLCFs of . Denote by . Let and represent its left coprime factors with inner. Assume that is an outer TFM such that there exists an outer TFM satisfying . is square and inner, it can be simply As proved that the Smith–McMillan form of can be expressed as with , in which is a monic polynomial, and the subscript is adopted to clarify that a TFM is related to a Smith–McMillan form. Moreover, let denote a zero of with multiplicity , then, it is . obvious that
Z
j
1In this paper, the -transform of a sequence a , denote it by a(z ), is a z , which is different from that of [16], [17] but defined as a(z ) = widely adopted in control theory. However, the parameterization of [17] is still z . valid after introducing a transformation z
!
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Using these symbols, a parameterization is derived in [17] for . all the NRCFs of TFMs internally stabilizable by is an outer TFM such that the Lemma 1: Assume that desirable TFM exists. Denote the th row of by . Then, and are NRCFs of a TFM internally , if and only if there exists a stable TFM , stabilized by and such that
criterion of minimizing the asymptotic bias and the asymptotic variance of the estimate. The first stage of the estimation is dealt with in the next section, while the second one in Section IV. III. FREQUENCY RESPONSE ESTIMATE FOR NRCFS AND ITS ASYMPTOTIC STATISTICAL PROPERTIES are When NRCFs of plants internally stabilizable by parameterized by (1), the relations can be expressed by the following equation, among the -transforms of the measured plant input/output, the probing signals, the external disturbances, and the measurement errors:
(1)
(2) whenever . , then, Moreover, when a plant is internally stabilizable by and an for every pair of its NRCFs, there exist an outer such that (1) and (2) are satisfied. inner In the interpolation conditions of (2), derivatives of are involved. This makes it not very convenient to be directly applied to our estimation problem. In [1] and [17], it is proved satisfying (2) and that the set consisting of all the stable can be expressed as
(4) in which Note that and troller. There exist stable TFMs (4) from the left by
. are NRCFs of the conand such that . Multiplying both sides of , we have
(3) in which termined by
-unitary TFM completely deand . As the actual construction , is not needed in the following estimation of algorithm, it is not included here. An interested reader can find it in [17]. Note that when a nonparametric estimate is required for the plant NRCFs, we are only interested in estimating their value . From Lemma 1 and (3), it can be further proved at that this estimation problem is equivalent to that of the inner . Moreover, in the latter estimation, it is not necesTFM sary to take into account the interpolation conditions in (2). This seems to be a significant advantage to divide NCFPMS identification into a nonparametric identification step and a frequency response estimate based parametric identification step. A detailed discussion on this issue, as well as a rigorous mathematical proof, are given in [17]. In the rest of this paper, NRCF nonparametric estimation is discussed on the basis of (1). As usual, is adopted to denote an estimate. Similar to the frequency-domain case, the estimation procedure is divided into two stages. At the first stage, a nonparametric estimate is derived for plant NRCFs under the condition that , as well as anand , are prescribed. other two auxiliary TFMs At the second stage, these TFMs are selected according to the
(5)
is a
Equation (5) suggests that identification of is in fact is an open-loop identification problem. Recalling that required to be unitary, this identification problem is a little different from conventional open-loop frequency response estimations. It is this slight difference that invalidates a direct adoption of the ratio between the discrete Fourier transforms (DFTs) of the plant inputs and the plant outputs as an estimate for the fre. quency response of On the basis of these observations, the following procedure is proposed for NRCF frequency response estimations. Nonparametric Estimation Algorithm Step 1) Let fine
denote the right shift operator. De, as follows: (6)
(7) (8)
ZHOU: FREQUENCY RESPONSE ESTIMATION FOR NCFS
Step 2) Denote the DFTs of respectively, by cost function minimizer of
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and , and . Define as . Let be the under the condition
. Step 3) Let
(9) In the previous estimation procedure, is generally not stable. However, is always stable from its construction. This implies that when the excitation signals are of finite magnitude, both and , are also of finite magnitude. Therefore, and are well defined. On the other hand, direct algebraic operations show that when has full-column rank, can be expressed as follows:
(10) is not of full column rank, an analytic When expression can also be derived for the optimal through singular value decomposition. In this case, however, the optimal is not unique, which means that the estimation problem is not well posed [8]. Therefore, in practical engineering, this case should be avoided as much as possible. Unlike conventional plant nonparametric estimation, the gain have been explicitly taken into account in restrictions on the aforementioned estimation procedure. It is these restrictions that guarantee the estimate to be consistent with the available a priori plant information. On the other hand, it is the Frobenius norm, rather than the maximal singular value, of a complex matrix, that is adopted in the cost function. Recall that in norm of a TFM is widely adopted in robust control theory, model uncertainty quantification which is induced from the norms of input/output signals. These imply that the cost funcis consistent with robust control theory. Furtion thermore, in different identification experiments and/or at different frequencies, the strengths of the external disturbances and the measurement errors may possibly be different. To reflect these differences in NRCF nonparametric estimation, it seems to be more appropriate to adjust the cost function , in which is a frequency weighting matrix. For notational simplicity, this general situation is not discussed in this paper, but similar results can be established through completely the same arguments. is always guaranteed to be unitary, and all the other As matrices at the right-hand side of (9) are with elements of finite magnitudes, it is apparent that both the mathematical mean and the covariance matrix of the estimate are also with bounded elements. This is significantly different from some closed-loop nonparametric estimation algorithms, in which the variance of
the estimate may be infinite [7], [15]. On the other hand, note , external disturbances and that in the complex matrix is a nonmeasurement errors are included. Moreover, linear matrix-valued function of in which the inverse of the square root of a positive–definite random matrix is involved. This nonlinear relation makes the analysis much more matheand matically involved for the statistical properties of , compared with those for a nonparametric estimate of the plant itself, or those for the DFT of a time series, or even those for the frequency-domain data based closed-loop NRCF nonparametric estimate [2], [8], [10], [11], [16]. In estimations, both the estimate and its quality evaluation are important. The remainder of this section is devoted to the analysis of the asymptotic statistical properties of the nonparametric NRCF estimate in (9). From the nonparametric estimation procedure, it is apparent is known, the NRCFs of the auxthat when controller iliary plant , the auxiliary TFMs and , as , are determined irrespective of well as the outer TFM identification experimental data. This implies that the statistical properties of the estimate are mainly determined by those of . Therefore, we at first discuss its asymptotic statistical properties. For this end, define , and , respectively, as follows:
(11)
(12) in which
. Note that
(13) and represent NLCFs of the plant. MoreHere, over, the internal stability of the closed-loop system in Fig. 1 is outer. Therefore, implies that from the results on time series analysis and nonparametric estimation [2], [8], [10], [11], it can be directly declared that and are weak stationary zero-mean random when sequences with finite second-order central moments, with converges with probaincreasing , every column of bility 1 (w.p. 1) to a zero-mean random vector with complex or normal distribution, except at the angular frequency .2 Moreover, these random vectors are independent over the angular frequency . 2At these angular frequencies, V (j! ) is generally a real-valued random matrix. However, under the same conditions as those for 0 < ! < , it can be proved similarly that when ! = 0 or ! = , every column of V (j! ) converges w.p. 1 to a normally distributed random vector with increasing N [2], [8], [10], [11]. In the remainder of this paper, most of the discussions are restricted to the case in which ! 2 (0; ), in order to avoid an awkward presentation. But the results are still valid when ! 2 [0; ], provided that a slight modification similar to the above statements is made.
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In the following discussions, it is assumed without loss and of generality that the probing signals , are of finite magnitude. At first, the following conclusions are established for the relaand , while their proofs are included tions between in the Appendix. Lemma 2: Assume that the external disturbances , as well as the measurement errors and , are of finite magnitude. Then, with increasing converges to . The . convergence rate is at least has full column rank, it is To guarantee that has full row rank. In other words, necessary that matrix must be right invertible in order to make the esmatrix timate of (10) valid. From (6) and (8), it is obvious that the can be easily guaranteed through right invertibility of and . an appropriate selection of the probing signals Moreover, a similar condition is generally requested in conventional nonparametric estimations, even when identification experiments are performed in open-loop [8], [10]. These imply is in general not restricthat the previous condition on tive and satisfying it is physically significant. Based on this observation, all the following discussions are made under the ashas full-row rank. Moreover, sumption that is denoted by for a concise presentation. With this symbol, the conclusions of Lemma . 2 can be expressed as On the basis of this result, the following conclusions can be obtained, which give some important asymptotic statistical . properties of the estimate is right invertible and there Theorem 1: Assume that exists a positive such that its smallest nonzero singular value increases at a speed with increasing . Then, when and are zero-mean weak stationary random sequences with is asymptotically limited second order central moments, unbiased and all of its columns are asymptotically with a com; with a normal distriplex normal distribution when or . bution when A proof of Theorem 1 is given in the Appendix. is only right invertible, by the same token as When that taken in the Proof of Theorem 1, we can derive the following conclusions. As the proof is quite similar, the details are omitted. has full-row rank. Then, Theorem 2: Assume that under the same conditions as those of Theorem 1 about the exconternal disturbances and the measurement errors, verges with increasing in distribution to
(14) in which is a random matrix with all its columns asymptotically having normal/com. Moreplex normal distributions when is asymptotically independent of whenover, . ever
and are outer. It is apparent that is also outer and therefore has no transmis. Define sion zeros on the unit circle . Then, on the basis of the fact that , using similar arguments as those in the proof of Lemma 2, it can be shown that with increasing converges to with a convergence rate not smaller than . On the other hand, when the excitation signal , is periodic, it is well known that at the frequencies of the excitation signals, the smallest nonzero increases at a rate of . However, singular value of , is a realizawhen the probing signal tion of some stochastic sequences, for example, the extensively utilized pseudo-random binary signals, the nonzero singular generally fluctuate around the spectra of the values of probing signals which are usually bounded [2], [8], [10], [11]. This implies that when the probing signals are carefully chosen, the conditions of Theorem 1 can be satisfied. and are linear functions of As both , the statistical properties of the former can be easily obtained from those of the latter. In fact, their distribution functions apparently have similar forms. But it is worthwhile is only a necessary to emphasize that the consistency of and . This point is condition for those of made clear by Theorem 3 in the next section. Recall that
IV. AUXILIARY TFM SELECTION In the previous section, a procedure is suggested for estimating the frequency response of the plant NRCFs. In this estimation, some auxiliary TFMs and some factorizations are required. However, their influences on the estimation quality are not clear. This is an important issue in the nonparametric estimation, as these TFMs and factorizations are generally not unique [9], [13], [16], [17]. From linear system theory [9], [13], it can be declared that all the factorizations involved in the nonparametric NRCF estimation are unique up to orthogonal matrices. On the basis of this observation, through a direct but careful check of the nonparametric estimation procedure, it can be shown that the statistical properties of the estimate are independent of a particular choice of the factors and . This feature of the closed-loop time-domain NRCF nonparametric estimation algorithm is completely the same as that of its frequency-domain counterpart [16], [17]. As the proof is quite direct and obvious, the details are omitted. Therefore, a really essential issue in closed-loop time-domain NRCF nonparametric estimation is about the appropriate choice of the auxand . iliary TFMs Recall that in quality evaluation for an estimation algorithm, estimation bias and variance are widely employed. Moreover, a consistent estimate with minimal variance is generally appreciated [2], [8], [10], [11]. In the remainder of this section, we discuss the selection of the auxiliary TFMs under the criteria that the nonparametric NRCF estimate is asymptotically unbiased and its variance is asymptotically optimal.
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To settle this problem, let and repreas sent actual plant NRCFs, and define . Then, from the structure of the closed-loop system, it is not difficult to prove that
While the condition in Theorem 3 seems to be a great impediment to NRCF identification, it is relatively easy to have . Note that from Fig. 1, it can be directly an estimate for and to , proved that the TFM from , has the following expression: denote it by (16)
(15) From these relations, conditions can be established for the asymptotic unbiasedness of the estimate. Theorem 3: The nonparametric estimate of (9) is asymptotically unbiased only if . Moreover, if and matrix is right invertible while with its smallest nonzero singular value increases at a speed , then, the estimate is certainly asymptotically unbiased. A proof of Theorem 3 can be found in the Appendix. From this theorem, it is obvious that in order to get an asymptotically unbiased nonparametric NRCF estimate, some knowledge on the NRCFs is required. This is in agreement with the results on SISO systems and frequency-domain data based estimations. There are, however, some slight differences in the conditions between the frequency- and the time-domain data based estimations. In detail, the satisfaction of at all the angular frequencies to be estimated, denote them by , is a necessary and sufficient condition for the estimate to be correct when frequency-domain experimental data are available. But is required to guarantee the consistency of the estimate when time-domain experimental data are available. Obviously, the latter condition is stronger than its former counterpart. However, when there is a great amount of frequencies at which the frequency response of plant NRCFs is asked to be estimated, it can be simply proved on the basis of interpolation theory that the differences between these two conditions are not very significant [1], [16], [17]. However, it is worthwhile to mention that the construcand are quite different tions of the desirable between time- and frequency-domain closed-loop nonparais. These metric NRCF estimations, no matter how large differences result from the boundary Nevanlinna–Pick interpolation theory. More specifically, concerning the condition , the simultaneous satisfaction at all the required angular frequencies and the independent satisfaction at every individual required angular frequency are equivalent to each other. Note that when frequency-domain experimental data are available, rather than and , it is and that are needed in the closed-loop nonparametric NRCF estimation. These imply that the frequency-domain estimation problem can be settled by means of constructing the desirable and through direct and simple matrix operations, no matter whether the plant is of single-input–single-output (SISO) or of MIMO [16], [17]. But it is much more mathematand such ically involved to construct a pair of is satisfied. This becomes clear in that Lemma 3 and Theorem 4.
Recalling that and shown that
and are NRCFs of the plant, while NLCFs of the controller, it can be further
(17) On the other hand, the internal stability of the closed-loop system implies that is square and outer. As can be easily estimated from closed-loop experimental data using open-loop identification methods, these relations mean that is known, can be identified. If is when and available, it is possible to construct desirable such that . The details are given in Lemma 3 and Theorem 4. It is worthwhile to point out that even when an estimate for is available, can be identified only unique up to an orthogonal matrix. This nonuniqueness, however, does not cause any problems in NRCF estimation, recalling that NRCFs have the same properties. When all the conditions in Theorem 3 except are satisfied, through similar arguments as those in the proof of Theorem 1, it can be shown that with increasing , the estimation bias converges to a constant, while the estimation variance decrease monotonically to zero. The conclusions of Theorem 3 can be further enhanced so that the stochastic properties of the nonparametric estimation are and . Therecompletely determined by fore, the selection of the auxiliary plant itself is not essential. The proof is similar to that of the frequency-domain aprroach [16], [17], and is not included here for space considerations. Now, we discuss how to construct a pair of desirable and . To achieve this objective, the following results are at first established. and are NRCFs of a plant inLemma 3: and satisfy ternally stabilizable by , if and only if they can be expressed as
(18) in which is a TFM satisfying is a stable TFM satisfying . Assume that . Here,
and
are respectively
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inner and outer TFMs, and right coprime with
and by
directly shown that
are . Denote
. Then, it can be and
(19) Therefore, a necessary condition for the stability of both and in (18) is that is stable. This further implies . that Denote the Smith–McMillan form of by , in which . As is inner, this decomposition is always possible. Moreover, it is that also apparent from the inner property of . Define to be the th column of . Then, using similar arguments as those in the proof of Lemma 1 [17], the following conclusions can be proved. and of (18) are stable, if and Theorem 4: is stable and for every only if
(20)
On the basis of Lemma 3 and Theorem 4, all the desirable and can be parameterized by an LFT of an inner TFM. The parameterization and its derivations are quite similar to those of (3), and are therefore omitted. An interested reader is referred to [1], [16], and [17]. and and are also In addition to required in the suggested nonparametric NRCF estimation prois satisfied by the cedure. Note that when NRCFs of the auxiliary plant , it is apparent from (46) and Theorem 2 that the asymptotic statistical properties of the esti. mate is completely determined by the random matrix Moreover, the smaller the covariance matrix of each column of , the better the estimation quality is. Therefore, a physiand is to select those cally significant choice for that minimize the maximal eigenvalue of this covariance matrix. To have an appropriate choice for these stable TFMs, let and represent respectively the DFTs and . Then, from the definitions of of and , the stabilities of the closed-loop system and , as well as the reand the TFMs is inner, it can be easily proved that when quirement that the external disturbances and the measurement errors are of finite magnitude
(21)
. Here, is and the convergence rate is at least defined as , and are defined in a similar way. To simplify discussions, it is assumed afterwards, without too much loss of generality, that for every . Moreover, the external disturbances and the measurement errors of different identification experiments are assumed to be independent stands for the set of sequences of each other. Here, of complex random vectors which converges in distribution to the complex normal distribution with mean and covariance matrix . These assumptions are quite reasonable when and the environments have not changed significantly during identification experiments [2], [8], [10], [11]. In this case, it is apparent from the previous equation that all the are independent of each other. Moreover, if columns of as we define matrix
(22) Then, it is not very hard to see that every column of is asymptotically with the complex normal distribution . From the previous discussions, it is obvious that a and is physically significant selection of based on the minimization of the maximal eigenvalue of . As both and are unitary, it is obvious that the sets consisting of the eigenvalues of and those of are equal to each other. Hence, it is and such that the maxappropriate to select is minimized. This selection can imal eigenvalue of be achieved through a similar approach as that in the frequency-domain SISO system case [16]. As the arguments are quite similar and there are almost no technical difficulties in extending the results on SISO systems to MIMO systems, the details are not included. On the other hand, in practical nonparametric estimation, is available, and the actual is only an estimate for generally not known. In this case, although the first equality of (46) remains valid, the second one is only valid approximately. In order to reduce the estimation bias, it is obvious that the difference between the left- and the right-hand sides of the second equality in this equation should be made as small as with possible. Assume that , in which is an outer TFM representing estithe frequency characteristics of the error bound in and are chosen to satisfy mation. Then, when , it can be directly proved from Lemma 3 that . Moreover
(23)
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From this equality and the left-hand side of the second equality of (46), it is apparent that another physically signifiand is to let them cant selection for the auxiliary satisfy and minimize the norm of
for all the stable with norm not greater than 1. Apparently, this requirement is equal to the minimization of the norm of , which can be achieved through the same procedure as that for the minimiza. tion of The last TFM required to be determined before the nonpara. From Lemma 3, it metric estimation is the outer TFM can be proved through direct algebraic manipulations that when is satisfied, . Assume that is an outer TFM satis. Then, can be fying . Therefore, expressed as . Moreover, direct algebraic operations show that
its stabilizing controller are designated to be respectively as follows, which are the same as those adopted in [17]:
The measurement errors and , as well as the external dis, are assumed to be independent zero-mean white turbance Gaussian stationary random sequences. Their covariance ma, and . Six identifitrices are respectively cation experiments have been performed to get an estimate for the frequency response of the plant NRCFs. The probing signals are, respectively, as follows:
(24)
(25) and in (6) From these relations and the definitions of and (7), it can be claimed that the frequency response estimate for plant NRCFs, as well as its statistical properties, are inde. Therefore, except to be outer and to guarantee pendent of in Lemma 3, can the existence of the desirable TFM be selected arbitrarily. V. NUMERICAL SIMULATION EXAMPLE In this section, some numerical simulation results are reported to illustrate the proposed nonparametric estimation procedure. The simulation results also provide some confirmations on the derived theoretical results. In these simulations, the plant and
Two types of identification experiments are designed. In the first set, and are chosen as . In the and are chosen to be independent pseudosecond set, random binary signals generated by the Matlab file idinput.m with the default parameters. The produced signals are constant and a over intervals of length 1, with a periodicity of magnitude of 1. In frequency response estimations, is selected to be , while and , respectively, as shown in the equation at the bottom of the is chosen to be page. Moreover, the auxiliary plant , in which takes one of the following five TFMs:
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Fig. 2. Maximum singular value of Q (j! )
0Q
(j! ). :
R (z ); }: R (z );
with
Denote
with respect to by . In Fig.2, the maximum singular value of , which is called auxiliary plant NRCF mismatch at the angular frequency , is given. As NRCFs of a TFM are only unique up to an orthogonal matrix, which minimize in drawing this figure, the NRCFs of are utilized. When the probing signals are periodic, Fig. 3 shows the variations of the maximum singular value of the sample mean for the , nonparametric estimation error at the angular frequency as well as that of the sample covariance matrix. In this simulation, varies from 50 to 5000. Moreover, in order to obtain the sample mean and the sample covariance matrix, 100 trails are is logarithmically uniperformed at every . Furthermore, formly divided between 50 and 5000, and 50 samples have been calculated. From Fig. 3, it is clear that when the probing signals are periodic, the maximal singular value of the sample covariance matrix for the estimation error decreases almost in proportion to , no matter whether the condition is satisfied or not. Moreover, the estimation bias decreases monotonically with increasing , provided that is satisfied. When this condition is not satisfied, the estimation bias converges to a constant. It is also clear from these simulais from , the bigger the tions that the further magnitude of the constant is to which the estimation bias converges.
: R (z ) ;
4: R (z); and ?: R (z).
To investigate the influences of excitation signals on the estimation quality, similar simulations have been performed when the probing signals are pseudo-random binary signals. The results are presented in Fig. 4. Compared with those of Fig. 3, it is obvious that even when the condition is satisfied, the estimate is no longer asymptotically unbiased. Moreover, the maximal singular value of the estimation covariance matrix does not decrease monotonically with increasing . These results agree well with the theoretical conclusions. In these simulations, the sample mean and the sample covariance matrix of the estimation error are calculated as follows. and denote respectively the estimation Let and at the errors for the frequency response of in the th trial of identification experiangular frequency . Then, the sample mean of the estimaments, tion error, denote it by , is computed as
while its sample covariance matrix as
Numerical simulations have also been performed to investiand , on gate the influences of the auxiliary the asymptotic statistical properties of the nonparametric estimate. The observed phenomena agree well with the theoretical results. The details, however, are omitted, due to space considerations.
ZHOU: FREQUENCY RESPONSE ESTIMATION FOR NCFS
Fig. 3.
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Estimation bias and estimation variance for periodic probing signals. : R (z ); }: R (z );
VI. CONCLUDING REMARKS
In this paper, an estimate is derived for the frequency response of plant NRCFs, on the basis of closed-loop time-domain experimental data. The plant is MIMO, and may possibly be unstable. The derivations are based on constrained data-fitting and an LFT parameterization for all the NRCFs of plants that can be internally stabilized by a known controller. An appreciable feature of this estimate is that the estimation bias and the estimation variance are always finite. This is significantly different from
: R (z ); 4: R (z ); and ?: R (z ).
some closed-loop nonparametric estimates for the plant itself, and can be regarded as one of the benefits of adopting NCFs in closed-loop system identification. Under the condition that the external disturbances and the measurement errors are zero-mean weak stationary random sequences with finite second order central moments, some important statistical properties of the estimate are investigated. It is proved that when the probing signals are periodic and the NRCFs of the auxiliary plant are appropriately selected, the estimate is asymptotically unbiased and normally/complex normally distributed when the angular frequency belongs to
48
Fig. 4.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 1, JANUARY 2006
Estimation bias and estimation variance for pseudo-random binary probing signals. : R (z ); }: R (z );
/ . The asymptotic statistical properties of the estimate have also been discussed for general exciting signals. It has been made clear that the estimate can asymptotically be expressed by a linear function of a normalized random matrix with all its columns being a normal/complex normal random vector when the angular frequency takes a value from / . Influences have also been investigated for the auxiliary TFMs on the estimation quality. Moreover, a computationally tractable procedure is suggested for desirable auxiliary TFM selections, based on the criteria of asymptotic unbiasedness and asymptotically minimal estimation variance.
: R (z ); 4: R (z ); and ?: R (z ).
Some numerical simulation results have been reported which confirm the theoretical conclusions. APPENDIX THEOREM PROOFS Proof of Lemma 2: Note that both and are stable, is outer. From the definition of and the assumptions that the probing signals are of finite magnitude, it is obvious that every element of the matrix , is of finite magnitude. As matrix
ZHOU: FREQUENCY RESPONSE ESTIMATION FOR NCFS
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is with a finite dimension, we therefore can declare that is finite, . On the basis of similar and arguments, it can be claimed that when are of finite magnitude, is also . For notational simplicity, denote finite, and by and , respectively. and are stable, we On the other hand, as and can be respectively expressed as have that and with and , in which and are finite numbers. , it is apparent that From the definition of
This completes the proof. Proof of Theorem 1: In this proof, for notational simand are respectively abbreviated to plicity, and , while and respectively to and . When and the assumptions on and are satisfied, it is well known that every column of converges to a zero-mean random vector with a w.p.1 with increasing complex normal distribution that has a finite covariance matrix . On the [2], [8], [10], [11]. This implies that other hand, from the assumptions on , it is obvious that . Therefore (29) is unitary from the definition of , the As first equality of the above equation further implies that . Moreover, from , we have that (30)
(26) A direct result of this relation is
(31) Note that
(32) Therefore
(27) (33) Therefore, for arbitrary real
Hence, when function of , we have that
with
a positive scalar
(34) As
, we therefore have that . That is (35)
On the other hand, define matrix
(28)
as
. Here, is a unitary matrix satisfying . only depends on the dimension It is well known that of matrix , and is independent of its elements. Then, it is not difficult to see that every element of matrix is of finite magnitude. This implies that the maximal singular is also finite. Note that is value of matrix asymptotically with a complex normal distribution and is unitary. Moreover, from (34), we have that
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. and, hence, are also Therefore, asymptotically with a complex normal distribution. More precisely
Based on this equation and the normality and coprimeness of and , through similar arguments as those in the proof of Lemma 1 [17], it can be shown that when the desirable exists, there is certainly an inner satisfying
(36) stands for the covariance matrix of , and . . As Now, consider the case in which is always positive definite when is right invertible, it is not difficult to see that there exists a unique Hermitian matrix , such that . From this observation and (30) and (31), it can be directly proved that in
which
(45) Form (15) and (45), as well as the definition of , direct algebraic manipulations show that for every , the following relations can be established:
(37)
(38)
(46)
Based on these relations and the definition of matrix , it can be shown that the aforementioned conclusions remain valid . The arguments are very similar even when is met but more to those when the condition tedious. The details are, therefore, omitted. can be shown by the same The conclusions for token. This completes the proof. Proof of Theorem 3: Assume that the estimate is asymptotically unbiased. Then
is right invertible and its smallest nonzero Then, when singular value increases with increasing at a speed , it can be proved by the same token as that of Theorem 1 that (47) which implies that the estimate is consistent. This completes the proof. Proof of Lemma 3: When (18) is satisfied, it is obvious that (48)
(39) This is equivalent to
(49)
(40) Multiplying both sides of the aforementioned equation from the left by , and noting that , we have that
On the contrary, when have from the definitions of
and
This implies that there exists a stable TFM
As both and are outer according to their definitions, and are right coprime these relations imply that and can be internally stabilized by . Moreover, direct algebraic manipulations show that
(41)
(50)
is satisfied, we that
and are also normalized. Therefore, and are NRCFs On the contrary, assume that of a plant internally stabilizable by and satisfy . Then, both and are stable, and
(42)
(51)
such that
(52) (43)
is outer from its assumptions, the previous equation As satisfying further implies the existence of a stable TFM
(44)
as Define from (51), it can be easily proved that
. Then, (53)
Substituting this relation into (52), it can be further shown that (54)
ZHOU: FREQUENCY RESPONSE ESTIMATION FOR NCFS
This implies the existence of a TFM and
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satisfying
(55) Note that have
. We therefore
(56) From the previous relation and (53), it can be simply proved that
(57) This completes the proof. ACKNOWLEDGMENT The author would like to thank the associate editor and the reviewers for their constructive comments.
[7] W. P. Heath, “Probability density function of indirect nonparametric transfer function estimates for plants in closed-loop,” in Proc. 2000 SYSID, Santa Barbara, CA, 2000, pp. 321–326. [8] L. Ljung, System Identification: Theory for the User, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1999. [9] D. McFarlane and K. Glover, Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, ser. Lecture Notes in Control and Information Sciences. New York: Springer-Verlag, 1990, vol. 38. [10] P. Pintelon and G. Schoukens, System Identification: A Frequency Domain Approach. New York: IEEE Press, 2000. [11] M. B. Priestley, Spectral Analysis and Time Series. New York: Academic, 2001. [12] P. M. J. Van den Hof, R. J. P. Schrama, R. A. Callafon, and O. K. Bosgra, “Identification of normalized coprime factors from closed-loop experimental data,” Eur. J. Control, vol. 1, no. 1, pp. 62–74, 1995. [13] M. Vidyasagar, Control System Synthesis: A Factorization Approach. Cambridge, MA: MIT Press, 1985. [14] G. Vinnicombe, “Frequency domain uncertainty and the graph topology,” IEEE Trans. Autom. Control, vol. 38, no. 9, pp. 1371–1383, Sep. 1993. [15] J. S. Welsh and G. C. Goodwin, “Finite sample properties of indirect nonparametric closed-loop identification,” IEEE Trans. Autom. Control, vol. 47, no. 8, pp. 1277–1292, Aug. 2002. [16] T. Zhou, “On the computation and statistical properties of the MLE for the frequency response of NCFs,” Int. J. Control, vol. 76, no. 15, pp. 1547–1559, 2003. , “Nonparametric estimation for normalized coprime factors of a [17] MIMO system,” Automatica, vol. 41, no. 4, pp. 655–662, 2005. [18] T. Zhou and H. W. Xing, “Identification of normalized coprime factors through constrained curve fitting,” Automatica, vol. 40, no. 9, pp. 1591–1601, 2004.
REFERENCES [1] J. A. Ball, I. Gohberg, and L. Rodman, Interpolation of Rational Matrix Functions. Operator Theory: Advances and Applications. Basel, Germany: Birkhäuser, 1990, vol. 45. [2] D. R. Brillinger, Time Series: Data Analysis and Theory. San Francisco, CA: Holden-Day, 1981. [3] P. Data and G. Vinnicombe, “An algorithm for identification in the v-gap metric,” in Proc. 38th Conf. Decision and Control, Phoenix, AZ, 1999, pp. 3230–3235. [4] U. Forssell and L. Ljung, “Closed-loop identification revisted,” Automatica, vol. 35, no. 7, pp. 1215–1241, 1999. [5] T. Georgiou and M. C. Smith, “Optimal robustness in the gap metric,” IEEE Trans. Autom. Control, vol. 35, no. 5, pp. 673–686, May 1990. [6] G. X. Gu, “Modeling of normalized coprime factors with v -metric uncertainty,” IEEE Trans. Autom. Control, vol. 40, no. 8, pp. 1498–1511, Aug. 1999.
Tong Zhou (M’02) was born in Hunan Province, China, in 1964. He received the B.S. and M.S. degrees from the University of Electronic Science and Technology of China, Chengdu, China, in 1984 and 1989, respectively, and the Ph.D. degree from Osaka University, Osaka, Japan, in 1994. After visiting several universities in the Netherlands, China, and Japan, he joined Tsinghua University, Beijing, China, in 1999, where he is now a Professor of control theory and control engineering. His research interests include robust control, system identification, signal processing, hybrid systems, communication systems, and their applications to real-world problems. Dr. Zhou was a recipient of the first class natural science prize in 2003 from the Ministry of Education, China.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 1, JANUARY 2006
Frequency Response Estimation for NCFs of an MIMO Plant From Closed-Loop Time-Domain Experimental Data Tong Zhou, Member, IEEE
Abstract—Frequency response estimation from closed-loop time-domain experimental data is investigated in this paper for normalized coprime factors (NCF) of a multiple-input–multiple-output (MIMO) plant. Based on a linear fractional transformation (LFT) representation for all the NCFs of plants internally stabilizable by a known controller, this estimation problem is converted into the open-loop nonparametric estimation of an inner transfer function matrix (TFM). An estimate is derived through constrained data-matching. It is proved that when the probing signals are periodic and the NCFs of the auxiliary plant are appropriately selected, the estimate is asymptotically unbiased and with a normal/complex normal distribution. It has been made clear that the estimation bias and the estimation variance are always finite. Computationally tractable procedures are suggested for choosing the desirable auxiliary TFMs. Index Terms—Closed-loop identification, multiple-input–multiple-output (MIMO) system, normalized coprime factorization, robust control.
I. INTRODUCTION
I
N INDUSTRIAL applications, it is often required that system identification is performed in closed-loop due to safety and economy considerations, etc. While investigations on closed-loop system identification have a long history, some important issues still need further studies [7], [15], [4], [8]. For example, when an indirect approach is adopted in nonparametric estimation, the estimate is possibly with an infinite variance; when a direct approach is employed in deriving a parametric model, a noise model is required that can describe the actual disturbance characteristics, etc. In representing plant dynamics, many kinds of models have been suggested. Transfer functions and state space models are two instances that are extensively applied in system synthesis. Recently, NCF-based representations became popular, mainly due to its appreciable properties in representing the dynamics of unstable plants, its close relations with the gap metric and the -gap metric [13], [6], [5], [14], [9]. Although model sets with NCF perturbations (NCFPMS) are well utilized in robust control theory, it is not very easy to identify them from experimental data. While great advances have Manuscript received July 21, 2004; revised April 25, 2005 and July 29, 2005. Recommended by Associate Editor E. Bai. This work was supported in part by the National Natural Science Foundation of China under Grant 60174022, and in part by the Trans-Century Training Programme Foundation for the Talents of MOE, P.R.C., Tsinghua University under Grant JC2003058. The author is with the Department of Automation, Tsinghua University, Beijing 100084, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.860279
been made, respectively, in [3], [6], and [12] toward NCFPMS estimation, practical applications of the suggested algorithms can still be hardly regarded as very convenient [16]–[18]. An essential issue here seems to be the lack of mathematically appreciable relations between NCFs and physically measurable variables [9], [6], [3], [12]. To solve this identification problem, it is suggested in [16]–[18] to divide it into two steps. At first, frequency response is estimated for NCFs. Then, a parametric NCFPMS is identified. While theoretical and numerical studies show that this approach is really capable of providing a NCFPMS from experimental data, all the studies are based on frequency-domain experimental data. Moreover, in deriving the analytic expression for the nonparametric estimate, covariance matrices are required for external disturbances and measurement errors. There is, however, no guarantee that this information is always available in practical engineering [8], [10]. In this paper, we discuss time-domain experimental data based nonparametric NCF estimation for MIMO plants in a closed-loop system. For a concise presentation, only normalized right coprime factors (NRCF) are discussed. When plant normalized left coprime factors (NLCFs) are required to be identified, it can be dealt with through coping with the transpose of the plant. Based on an LFT parameterization for all the NRCFs of plants internally stabilizable by a known controller, closed-loop nonparametric NRCF estimation is converted into the open-loop nonparametric estimation for an inner TFM. A procedure is proposed for estimating the frequency response of plant NRCFs, through constrained data-fitting from a batch of closed-loop time-domain experimental data. It has been made clear that the estimation bias and variance are always finite, which is significantly different from nonparametric estimations for the plant itself. Some important statistical properties of the estimate are investigated under the condition that the external disturbances and the measurement errors are zero-mean weak stationary random sequences with finite second order central moments. It is proved that when the NRCFs of the auxiliary plant are appropriately selected and the probing signals are periodic, the estimate is asymptotically unbiased and normally/complex normally distributed. On the other hand, for general probing signals, the asymptotic statistical properties of the estimate are shown to be completely determined by a normalized random matrix with all its columns having normal/complex normal distributions. A computationally tractable method is suggested for choosing the desirable NRCFs of the auxiliary plant, as well as other involved auxiliary
0018-9286/$20.00 © 2006 IEEE
ZHOU: FREQUENCY RESPONSE ESTIMATION FOR NCFS
Fig. 1.
Closed-loop identification experiment.
TFMs. Some numerical simulation results are reported which confirm the theoretical results. It is worthwhile to point out that in both the frequencyand the time-domain data based closed-loop NRCF nonparametric estimations, the major issues are similar. The estimation algorithms, however, have completely distinctive features. Moreover, there are significant differences in overcoming the involved mathematical problems arising in estimation quality evaluations. Furthermore, some important theoretical conclusions, such as the construction and the selection condition for the desirable auxiliary TFMs, etc., are not completely the same. These differences and difficulties are mainly due to the following facts. When a frequency-domain approach is adopted, the experimental data at different sampled frequencies can generally be assumed independent and normally/complex normally distributed. This property enables us to derive a nonparametric NCF estimate using only the experimental data at the same sampled frequency. However, when a time-domain approach is employed, this assumption is typically no longer valid. The dependence among time-domain experimental data at different sampled time instants makes the analysis of the estimation algorithm much more mathematically involved. On the other hand, although the closed-loop nonparametric NRCF estimation problem has been converted into an open-loop one, the latter is not a trivial application of the existing results. The requirement that the related plant is inner invalidates the available nonparametric plant estimates. The rest of this paper is organized as follows. In the next section, the estimation problem is described and a parameterization is provided for all the NRCFs of plants internally stabilizable by a known controller. The nonparametric estimate is obtained in Section III, as well as its asymptotic statistical properties. The auxiliary TFM selection problem is dealt with in Section IV, while Section V reports some numerical simulation results. The paper is concluded by Section VI. Finally, an Appendix is included which gives proofs of the lemmas and theorems. The notation used in this paper is fairly standard. Its detailed definition is therefore not included, but can be found, for example, in [5], [9], [6], and [16]. II. PROBLEM DESCRIPTION AND PRELIMINARY RESULTS Typically, a closed-loop system used in identification has a structure like that depicted in Fig. 1. Here, and are probing signals designed to obtain some dynamical characterand are, respectively, measureistics of the plant; reprement errors of the plant input and output, while sents the composite influence of exterior disturbances on the closed-loop system. On the other hand, and represent, respectively, the plant and the controller. In this paper, we only
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consider linear time-invariant plants and controllers. It is asinputs sumed that both the plant and the controller are of and outputs, and identification experiments have been performed. Superscript is adopted to clarify that a variable or a function, etc., is related to the th identification experiment, . Moreover, we also assume that the controller is known and of finite McMillan degree. Let and , denote respectively the sampled values of the probing signals and plant input/output measurements in the th identification experiment. The objective is to derive an estimate for the frequency response of the plant NRCFs and analyze its statistical properties. Compared with the frequency-domain experimental data based estimation problem discussed in [16], [17], there are no assumptions in the problem formulation on the statistical properties of the external disturbances and the measurement errors. It will become clear afterwards (Theorems 1 and 3) that a consistent estimate can be derived under some weak conditions. Moreover, the assumptions adopted in the above problem description are quite standard in closed-loop system identification. On the other hand, it is worthwhile to mention that while frequency-domain experimental data are easier to be prefiltered and combined from different identification experiments, time-domain experimental data are generally much easier to be obtained [8], [10]. In this paper, only square plants are dealt with. The results, however, can be extended to nonsquare plants. Moreover, similar results can be derived through completely the same arguments for the nonparametric estimation of plant NLCFs. Similar to the frequency-domain data based closed-loop nonparametric NRCF estimate, a time-domain counterpart is derived in this paper also from a parameterization for all the NRCFs of plants that compose an internally stable closed-loop system with a known controller. To express this parameterizais employed which is internally tion, an auxiliary TFM and has and as its NRCFs.1 stabilizable by Moreover, to make statements concise, the following symbols stands for when are at first defined in which is real and rational. and and Assume that are, respectively, NRCFs and NLCFs of . Denote by . Let and represent its left coprime factors with inner. Assume that is an outer TFM such that there exists an outer TFM satisfying . is square and inner, it can be simply As proved that the Smith–McMillan form of can be expressed as with , in which is a monic polynomial, and the subscript is adopted to clarify that a TFM is related to a Smith–McMillan form. Moreover, let denote a zero of with multiplicity , then, it is . obvious that
Z
j
1In this paper, the -transform of a sequence a , denote it by a(z ), is a z , which is different from that of [16], [17] but defined as a(z ) = widely adopted in control theory. However, the parameterization of [17] is still z . valid after introducing a transformation z
!
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Using these symbols, a parameterization is derived in [17] for . all the NRCFs of TFMs internally stabilizable by is an outer TFM such that the Lemma 1: Assume that desirable TFM exists. Denote the th row of by . Then, and are NRCFs of a TFM internally , if and only if there exists a stable TFM , stabilized by and such that
criterion of minimizing the asymptotic bias and the asymptotic variance of the estimate. The first stage of the estimation is dealt with in the next section, while the second one in Section IV. III. FREQUENCY RESPONSE ESTIMATE FOR NRCFS AND ITS ASYMPTOTIC STATISTICAL PROPERTIES are When NRCFs of plants internally stabilizable by parameterized by (1), the relations can be expressed by the following equation, among the -transforms of the measured plant input/output, the probing signals, the external disturbances, and the measurement errors:
(1)
(2) whenever . , then, Moreover, when a plant is internally stabilizable by and an for every pair of its NRCFs, there exist an outer such that (1) and (2) are satisfied. inner In the interpolation conditions of (2), derivatives of are involved. This makes it not very convenient to be directly applied to our estimation problem. In [1] and [17], it is proved satisfying (2) and that the set consisting of all the stable can be expressed as
(4) in which Note that and troller. There exist stable TFMs (4) from the left by
. are NRCFs of the conand such that . Multiplying both sides of , we have
(3) in which termined by
-unitary TFM completely deand . As the actual construction , is not needed in the following estimation of algorithm, it is not included here. An interested reader can find it in [17]. Note that when a nonparametric estimate is required for the plant NRCFs, we are only interested in estimating their value . From Lemma 1 and (3), it can be further proved at that this estimation problem is equivalent to that of the inner . Moreover, in the latter estimation, it is not necesTFM sary to take into account the interpolation conditions in (2). This seems to be a significant advantage to divide NCFPMS identification into a nonparametric identification step and a frequency response estimate based parametric identification step. A detailed discussion on this issue, as well as a rigorous mathematical proof, are given in [17]. In the rest of this paper, NRCF nonparametric estimation is discussed on the basis of (1). As usual, is adopted to denote an estimate. Similar to the frequency-domain case, the estimation procedure is divided into two stages. At the first stage, a nonparametric estimate is derived for plant NRCFs under the condition that , as well as anand , are prescribed. other two auxiliary TFMs At the second stage, these TFMs are selected according to the
(5)
is a
Equation (5) suggests that identification of is in fact is an open-loop identification problem. Recalling that required to be unitary, this identification problem is a little different from conventional open-loop frequency response estimations. It is this slight difference that invalidates a direct adoption of the ratio between the discrete Fourier transforms (DFTs) of the plant inputs and the plant outputs as an estimate for the fre. quency response of On the basis of these observations, the following procedure is proposed for NRCF frequency response estimations. Nonparametric Estimation Algorithm Step 1) Let fine
denote the right shift operator. De, as follows: (6)
(7) (8)
ZHOU: FREQUENCY RESPONSE ESTIMATION FOR NCFS
Step 2) Denote the DFTs of respectively, by cost function minimizer of
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and , and . Define as . Let be the under the condition
. Step 3) Let
(9) In the previous estimation procedure, is generally not stable. However, is always stable from its construction. This implies that when the excitation signals are of finite magnitude, both and , are also of finite magnitude. Therefore, and are well defined. On the other hand, direct algebraic operations show that when has full-column rank, can be expressed as follows:
(10) is not of full column rank, an analytic When expression can also be derived for the optimal through singular value decomposition. In this case, however, the optimal is not unique, which means that the estimation problem is not well posed [8]. Therefore, in practical engineering, this case should be avoided as much as possible. Unlike conventional plant nonparametric estimation, the gain have been explicitly taken into account in restrictions on the aforementioned estimation procedure. It is these restrictions that guarantee the estimate to be consistent with the available a priori plant information. On the other hand, it is the Frobenius norm, rather than the maximal singular value, of a complex matrix, that is adopted in the cost function. Recall that in norm of a TFM is widely adopted in robust control theory, model uncertainty quantification which is induced from the norms of input/output signals. These imply that the cost funcis consistent with robust control theory. Furtion thermore, in different identification experiments and/or at different frequencies, the strengths of the external disturbances and the measurement errors may possibly be different. To reflect these differences in NRCF nonparametric estimation, it seems to be more appropriate to adjust the cost function , in which is a frequency weighting matrix. For notational simplicity, this general situation is not discussed in this paper, but similar results can be established through completely the same arguments. is always guaranteed to be unitary, and all the other As matrices at the right-hand side of (9) are with elements of finite magnitudes, it is apparent that both the mathematical mean and the covariance matrix of the estimate are also with bounded elements. This is significantly different from some closed-loop nonparametric estimation algorithms, in which the variance of
the estimate may be infinite [7], [15]. On the other hand, note , external disturbances and that in the complex matrix is a nonmeasurement errors are included. Moreover, linear matrix-valued function of in which the inverse of the square root of a positive–definite random matrix is involved. This nonlinear relation makes the analysis much more matheand matically involved for the statistical properties of , compared with those for a nonparametric estimate of the plant itself, or those for the DFT of a time series, or even those for the frequency-domain data based closed-loop NRCF nonparametric estimate [2], [8], [10], [11], [16]. In estimations, both the estimate and its quality evaluation are important. The remainder of this section is devoted to the analysis of the asymptotic statistical properties of the nonparametric NRCF estimate in (9). From the nonparametric estimation procedure, it is apparent is known, the NRCFs of the auxthat when controller iliary plant , the auxiliary TFMs and , as , are determined irrespective of well as the outer TFM identification experimental data. This implies that the statistical properties of the estimate are mainly determined by those of . Therefore, we at first discuss its asymptotic statistical properties. For this end, define , and , respectively, as follows:
(11)
(12) in which
. Note that
(13) and represent NLCFs of the plant. MoreHere, over, the internal stability of the closed-loop system in Fig. 1 is outer. Therefore, implies that from the results on time series analysis and nonparametric estimation [2], [8], [10], [11], it can be directly declared that and are weak stationary zero-mean random when sequences with finite second-order central moments, with converges with probaincreasing , every column of bility 1 (w.p. 1) to a zero-mean random vector with complex or normal distribution, except at the angular frequency .2 Moreover, these random vectors are independent over the angular frequency . 2At these angular frequencies, V (j! ) is generally a real-valued random matrix. However, under the same conditions as those for 0 < ! < , it can be proved similarly that when ! = 0 or ! = , every column of V (j! ) converges w.p. 1 to a normally distributed random vector with increasing N [2], [8], [10], [11]. In the remainder of this paper, most of the discussions are restricted to the case in which ! 2 (0; ), in order to avoid an awkward presentation. But the results are still valid when ! 2 [0; ], provided that a slight modification similar to the above statements is made.
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In the following discussions, it is assumed without loss and of generality that the probing signals , are of finite magnitude. At first, the following conclusions are established for the relaand , while their proofs are included tions between in the Appendix. Lemma 2: Assume that the external disturbances , as well as the measurement errors and , are of finite magnitude. Then, with increasing converges to . The . convergence rate is at least has full column rank, it is To guarantee that has full row rank. In other words, necessary that matrix must be right invertible in order to make the esmatrix timate of (10) valid. From (6) and (8), it is obvious that the can be easily guaranteed through right invertibility of and . an appropriate selection of the probing signals Moreover, a similar condition is generally requested in conventional nonparametric estimations, even when identification experiments are performed in open-loop [8], [10]. These imply is in general not restricthat the previous condition on tive and satisfying it is physically significant. Based on this observation, all the following discussions are made under the ashas full-row rank. Moreover, sumption that is denoted by for a concise presentation. With this symbol, the conclusions of Lemma . 2 can be expressed as On the basis of this result, the following conclusions can be obtained, which give some important asymptotic statistical . properties of the estimate is right invertible and there Theorem 1: Assume that exists a positive such that its smallest nonzero singular value increases at a speed with increasing . Then, when and are zero-mean weak stationary random sequences with is asymptotically limited second order central moments, unbiased and all of its columns are asymptotically with a com; with a normal distriplex normal distribution when or . bution when A proof of Theorem 1 is given in the Appendix. is only right invertible, by the same token as When that taken in the Proof of Theorem 1, we can derive the following conclusions. As the proof is quite similar, the details are omitted. has full-row rank. Then, Theorem 2: Assume that under the same conditions as those of Theorem 1 about the exconternal disturbances and the measurement errors, verges with increasing in distribution to
(14) in which is a random matrix with all its columns asymptotically having normal/com. Moreplex normal distributions when is asymptotically independent of whenover, . ever
and are outer. It is apparent that is also outer and therefore has no transmis. Define sion zeros on the unit circle . Then, on the basis of the fact that , using similar arguments as those in the proof of Lemma 2, it can be shown that with increasing converges to with a convergence rate not smaller than . On the other hand, when the excitation signal , is periodic, it is well known that at the frequencies of the excitation signals, the smallest nonzero increases at a rate of . However, singular value of , is a realizawhen the probing signal tion of some stochastic sequences, for example, the extensively utilized pseudo-random binary signals, the nonzero singular generally fluctuate around the spectra of the values of probing signals which are usually bounded [2], [8], [10], [11]. This implies that when the probing signals are carefully chosen, the conditions of Theorem 1 can be satisfied. and are linear functions of As both , the statistical properties of the former can be easily obtained from those of the latter. In fact, their distribution functions apparently have similar forms. But it is worthwhile is only a necessary to emphasize that the consistency of and . This point is condition for those of made clear by Theorem 3 in the next section. Recall that
IV. AUXILIARY TFM SELECTION In the previous section, a procedure is suggested for estimating the frequency response of the plant NRCFs. In this estimation, some auxiliary TFMs and some factorizations are required. However, their influences on the estimation quality are not clear. This is an important issue in the nonparametric estimation, as these TFMs and factorizations are generally not unique [9], [13], [16], [17]. From linear system theory [9], [13], it can be declared that all the factorizations involved in the nonparametric NRCF estimation are unique up to orthogonal matrices. On the basis of this observation, through a direct but careful check of the nonparametric estimation procedure, it can be shown that the statistical properties of the estimate are independent of a particular choice of the factors and . This feature of the closed-loop time-domain NRCF nonparametric estimation algorithm is completely the same as that of its frequency-domain counterpart [16], [17]. As the proof is quite direct and obvious, the details are omitted. Therefore, a really essential issue in closed-loop time-domain NRCF nonparametric estimation is about the appropriate choice of the auxand . iliary TFMs Recall that in quality evaluation for an estimation algorithm, estimation bias and variance are widely employed. Moreover, a consistent estimate with minimal variance is generally appreciated [2], [8], [10], [11]. In the remainder of this section, we discuss the selection of the auxiliary TFMs under the criteria that the nonparametric NRCF estimate is asymptotically unbiased and its variance is asymptotically optimal.
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To settle this problem, let and repreas sent actual plant NRCFs, and define . Then, from the structure of the closed-loop system, it is not difficult to prove that
While the condition in Theorem 3 seems to be a great impediment to NRCF identification, it is relatively easy to have . Note that from Fig. 1, it can be directly an estimate for and to , proved that the TFM from , has the following expression: denote it by (16)
(15) From these relations, conditions can be established for the asymptotic unbiasedness of the estimate. Theorem 3: The nonparametric estimate of (9) is asymptotically unbiased only if . Moreover, if and matrix is right invertible while with its smallest nonzero singular value increases at a speed , then, the estimate is certainly asymptotically unbiased. A proof of Theorem 3 can be found in the Appendix. From this theorem, it is obvious that in order to get an asymptotically unbiased nonparametric NRCF estimate, some knowledge on the NRCFs is required. This is in agreement with the results on SISO systems and frequency-domain data based estimations. There are, however, some slight differences in the conditions between the frequency- and the time-domain data based estimations. In detail, the satisfaction of at all the angular frequencies to be estimated, denote them by , is a necessary and sufficient condition for the estimate to be correct when frequency-domain experimental data are available. But is required to guarantee the consistency of the estimate when time-domain experimental data are available. Obviously, the latter condition is stronger than its former counterpart. However, when there is a great amount of frequencies at which the frequency response of plant NRCFs is asked to be estimated, it can be simply proved on the basis of interpolation theory that the differences between these two conditions are not very significant [1], [16], [17]. However, it is worthwhile to mention that the construcand are quite different tions of the desirable between time- and frequency-domain closed-loop nonparais. These metric NRCF estimations, no matter how large differences result from the boundary Nevanlinna–Pick interpolation theory. More specifically, concerning the condition , the simultaneous satisfaction at all the required angular frequencies and the independent satisfaction at every individual required angular frequency are equivalent to each other. Note that when frequency-domain experimental data are available, rather than and , it is and that are needed in the closed-loop nonparametric NRCF estimation. These imply that the frequency-domain estimation problem can be settled by means of constructing the desirable and through direct and simple matrix operations, no matter whether the plant is of single-input–single-output (SISO) or of MIMO [16], [17]. But it is much more mathematand such ically involved to construct a pair of is satisfied. This becomes clear in that Lemma 3 and Theorem 4.
Recalling that and shown that
and are NRCFs of the plant, while NLCFs of the controller, it can be further
(17) On the other hand, the internal stability of the closed-loop system implies that is square and outer. As can be easily estimated from closed-loop experimental data using open-loop identification methods, these relations mean that is known, can be identified. If is when and available, it is possible to construct desirable such that . The details are given in Lemma 3 and Theorem 4. It is worthwhile to point out that even when an estimate for is available, can be identified only unique up to an orthogonal matrix. This nonuniqueness, however, does not cause any problems in NRCF estimation, recalling that NRCFs have the same properties. When all the conditions in Theorem 3 except are satisfied, through similar arguments as those in the proof of Theorem 1, it can be shown that with increasing , the estimation bias converges to a constant, while the estimation variance decrease monotonically to zero. The conclusions of Theorem 3 can be further enhanced so that the stochastic properties of the nonparametric estimation are and . Therecompletely determined by fore, the selection of the auxiliary plant itself is not essential. The proof is similar to that of the frequency-domain aprroach [16], [17], and is not included here for space considerations. Now, we discuss how to construct a pair of desirable and . To achieve this objective, the following results are at first established. and are NRCFs of a plant inLemma 3: and satisfy ternally stabilizable by , if and only if they can be expressed as
(18) in which is a TFM satisfying is a stable TFM satisfying . Assume that . Here,
and
are respectively
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inner and outer TFMs, and right coprime with
and by
directly shown that
are . Denote
. Then, it can be and
(19) Therefore, a necessary condition for the stability of both and in (18) is that is stable. This further implies . that Denote the Smith–McMillan form of by , in which . As is inner, this decomposition is always possible. Moreover, it is that also apparent from the inner property of . Define to be the th column of . Then, using similar arguments as those in the proof of Lemma 1 [17], the following conclusions can be proved. and of (18) are stable, if and Theorem 4: is stable and for every only if
(20)
On the basis of Lemma 3 and Theorem 4, all the desirable and can be parameterized by an LFT of an inner TFM. The parameterization and its derivations are quite similar to those of (3), and are therefore omitted. An interested reader is referred to [1], [16], and [17]. and and are also In addition to required in the suggested nonparametric NRCF estimation prois satisfied by the cedure. Note that when NRCFs of the auxiliary plant , it is apparent from (46) and Theorem 2 that the asymptotic statistical properties of the esti. mate is completely determined by the random matrix Moreover, the smaller the covariance matrix of each column of , the better the estimation quality is. Therefore, a physiand is to select those cally significant choice for that minimize the maximal eigenvalue of this covariance matrix. To have an appropriate choice for these stable TFMs, let and represent respectively the DFTs and . Then, from the definitions of of and , the stabilities of the closed-loop system and , as well as the reand the TFMs is inner, it can be easily proved that when quirement that the external disturbances and the measurement errors are of finite magnitude
(21)
. Here, is and the convergence rate is at least defined as , and are defined in a similar way. To simplify discussions, it is assumed afterwards, without too much loss of generality, that for every . Moreover, the external disturbances and the measurement errors of different identification experiments are assumed to be independent stands for the set of sequences of each other. Here, of complex random vectors which converges in distribution to the complex normal distribution with mean and covariance matrix . These assumptions are quite reasonable when and the environments have not changed significantly during identification experiments [2], [8], [10], [11]. In this case, it is apparent from the previous equation that all the are independent of each other. Moreover, if columns of as we define matrix
(22) Then, it is not very hard to see that every column of is asymptotically with the complex normal distribution . From the previous discussions, it is obvious that a and is physically significant selection of based on the minimization of the maximal eigenvalue of . As both and are unitary, it is obvious that the sets consisting of the eigenvalues of and those of are equal to each other. Hence, it is and such that the maxappropriate to select is minimized. This selection can imal eigenvalue of be achieved through a similar approach as that in the frequency-domain SISO system case [16]. As the arguments are quite similar and there are almost no technical difficulties in extending the results on SISO systems to MIMO systems, the details are not included. On the other hand, in practical nonparametric estimation, is available, and the actual is only an estimate for generally not known. In this case, although the first equality of (46) remains valid, the second one is only valid approximately. In order to reduce the estimation bias, it is obvious that the difference between the left- and the right-hand sides of the second equality in this equation should be made as small as with possible. Assume that , in which is an outer TFM representing estithe frequency characteristics of the error bound in and are chosen to satisfy mation. Then, when , it can be directly proved from Lemma 3 that . Moreover
(23)
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From this equality and the left-hand side of the second equality of (46), it is apparent that another physically signifiand is to let them cant selection for the auxiliary satisfy and minimize the norm of
for all the stable with norm not greater than 1. Apparently, this requirement is equal to the minimization of the norm of , which can be achieved through the same procedure as that for the minimiza. tion of The last TFM required to be determined before the nonpara. From Lemma 3, it metric estimation is the outer TFM can be proved through direct algebraic manipulations that when is satisfied, . Assume that is an outer TFM satis. Then, can be fying . Therefore, expressed as . Moreover, direct algebraic operations show that
its stabilizing controller are designated to be respectively as follows, which are the same as those adopted in [17]:
The measurement errors and , as well as the external dis, are assumed to be independent zero-mean white turbance Gaussian stationary random sequences. Their covariance ma, and . Six identifitrices are respectively cation experiments have been performed to get an estimate for the frequency response of the plant NRCFs. The probing signals are, respectively, as follows:
(24)
(25) and in (6) From these relations and the definitions of and (7), it can be claimed that the frequency response estimate for plant NRCFs, as well as its statistical properties, are inde. Therefore, except to be outer and to guarantee pendent of in Lemma 3, can the existence of the desirable TFM be selected arbitrarily. V. NUMERICAL SIMULATION EXAMPLE In this section, some numerical simulation results are reported to illustrate the proposed nonparametric estimation procedure. The simulation results also provide some confirmations on the derived theoretical results. In these simulations, the plant and
Two types of identification experiments are designed. In the first set, and are chosen as . In the and are chosen to be independent pseudosecond set, random binary signals generated by the Matlab file idinput.m with the default parameters. The produced signals are constant and a over intervals of length 1, with a periodicity of magnitude of 1. In frequency response estimations, is selected to be , while and , respectively, as shown in the equation at the bottom of the is chosen to be page. Moreover, the auxiliary plant , in which takes one of the following five TFMs:
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Fig. 2. Maximum singular value of Q (j! )
0Q
(j! ). :
R (z ); }: R (z );
with
Denote
with respect to by . In Fig.2, the maximum singular value of , which is called auxiliary plant NRCF mismatch at the angular frequency , is given. As NRCFs of a TFM are only unique up to an orthogonal matrix, which minimize in drawing this figure, the NRCFs of are utilized. When the probing signals are periodic, Fig. 3 shows the variations of the maximum singular value of the sample mean for the , nonparametric estimation error at the angular frequency as well as that of the sample covariance matrix. In this simulation, varies from 50 to 5000. Moreover, in order to obtain the sample mean and the sample covariance matrix, 100 trails are is logarithmically uniperformed at every . Furthermore, formly divided between 50 and 5000, and 50 samples have been calculated. From Fig. 3, it is clear that when the probing signals are periodic, the maximal singular value of the sample covariance matrix for the estimation error decreases almost in proportion to , no matter whether the condition is satisfied or not. Moreover, the estimation bias decreases monotonically with increasing , provided that is satisfied. When this condition is not satisfied, the estimation bias converges to a constant. It is also clear from these simulais from , the bigger the tions that the further magnitude of the constant is to which the estimation bias converges.
: R (z ) ;
4: R (z); and ?: R (z).
To investigate the influences of excitation signals on the estimation quality, similar simulations have been performed when the probing signals are pseudo-random binary signals. The results are presented in Fig. 4. Compared with those of Fig. 3, it is obvious that even when the condition is satisfied, the estimate is no longer asymptotically unbiased. Moreover, the maximal singular value of the estimation covariance matrix does not decrease monotonically with increasing . These results agree well with the theoretical conclusions. In these simulations, the sample mean and the sample covariance matrix of the estimation error are calculated as follows. and denote respectively the estimation Let and at the errors for the frequency response of in the th trial of identification experiangular frequency . Then, the sample mean of the estimaments, tion error, denote it by , is computed as
while its sample covariance matrix as
Numerical simulations have also been performed to investiand , on gate the influences of the auxiliary the asymptotic statistical properties of the nonparametric estimate. The observed phenomena agree well with the theoretical results. The details, however, are omitted, due to space considerations.
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Fig. 3.
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Estimation bias and estimation variance for periodic probing signals. : R (z ); }: R (z );
VI. CONCLUDING REMARKS
In this paper, an estimate is derived for the frequency response of plant NRCFs, on the basis of closed-loop time-domain experimental data. The plant is MIMO, and may possibly be unstable. The derivations are based on constrained data-fitting and an LFT parameterization for all the NRCFs of plants that can be internally stabilized by a known controller. An appreciable feature of this estimate is that the estimation bias and the estimation variance are always finite. This is significantly different from
: R (z ); 4: R (z ); and ?: R (z ).
some closed-loop nonparametric estimates for the plant itself, and can be regarded as one of the benefits of adopting NCFs in closed-loop system identification. Under the condition that the external disturbances and the measurement errors are zero-mean weak stationary random sequences with finite second order central moments, some important statistical properties of the estimate are investigated. It is proved that when the probing signals are periodic and the NRCFs of the auxiliary plant are appropriately selected, the estimate is asymptotically unbiased and normally/complex normally distributed when the angular frequency belongs to
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Fig. 4.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 1, JANUARY 2006
Estimation bias and estimation variance for pseudo-random binary probing signals. : R (z ); }: R (z );
/ . The asymptotic statistical properties of the estimate have also been discussed for general exciting signals. It has been made clear that the estimate can asymptotically be expressed by a linear function of a normalized random matrix with all its columns being a normal/complex normal random vector when the angular frequency takes a value from / . Influences have also been investigated for the auxiliary TFMs on the estimation quality. Moreover, a computationally tractable procedure is suggested for desirable auxiliary TFM selections, based on the criteria of asymptotic unbiasedness and asymptotically minimal estimation variance.
: R (z ); 4: R (z ); and ?: R (z ).
Some numerical simulation results have been reported which confirm the theoretical conclusions. APPENDIX THEOREM PROOFS Proof of Lemma 2: Note that both and are stable, is outer. From the definition of and the assumptions that the probing signals are of finite magnitude, it is obvious that every element of the matrix , is of finite magnitude. As matrix
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is with a finite dimension, we therefore can declare that is finite, . On the basis of similar and arguments, it can be claimed that when are of finite magnitude, is also . For notational simplicity, denote finite, and by and , respectively. and are stable, we On the other hand, as and can be respectively expressed as have that and with and , in which and are finite numbers. , it is apparent that From the definition of
This completes the proof. Proof of Theorem 1: In this proof, for notational simand are respectively abbreviated to plicity, and , while and respectively to and . When and the assumptions on and are satisfied, it is well known that every column of converges to a zero-mean random vector with a w.p.1 with increasing complex normal distribution that has a finite covariance matrix . On the [2], [8], [10], [11]. This implies that other hand, from the assumptions on , it is obvious that . Therefore (29) is unitary from the definition of , the As first equality of the above equation further implies that . Moreover, from , we have that (30)
(26) A direct result of this relation is
(31) Note that
(32) Therefore
(27) (33) Therefore, for arbitrary real
Hence, when function of , we have that
with
a positive scalar
(34) As
, we therefore have that . That is (35)
On the other hand, define matrix
(28)
as
. Here, is a unitary matrix satisfying . only depends on the dimension It is well known that of matrix , and is independent of its elements. Then, it is not difficult to see that every element of matrix is of finite magnitude. This implies that the maximal singular is also finite. Note that is value of matrix asymptotically with a complex normal distribution and is unitary. Moreover, from (34), we have that
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. and, hence, are also Therefore, asymptotically with a complex normal distribution. More precisely
Based on this equation and the normality and coprimeness of and , through similar arguments as those in the proof of Lemma 1 [17], it can be shown that when the desirable exists, there is certainly an inner satisfying
(36) stands for the covariance matrix of , and . . As Now, consider the case in which is always positive definite when is right invertible, it is not difficult to see that there exists a unique Hermitian matrix , such that . From this observation and (30) and (31), it can be directly proved that in
which
(45) Form (15) and (45), as well as the definition of , direct algebraic manipulations show that for every , the following relations can be established:
(37)
(38)
(46)
Based on these relations and the definition of matrix , it can be shown that the aforementioned conclusions remain valid . The arguments are very similar even when is met but more to those when the condition tedious. The details are, therefore, omitted. can be shown by the same The conclusions for token. This completes the proof. Proof of Theorem 3: Assume that the estimate is asymptotically unbiased. Then
is right invertible and its smallest nonzero Then, when singular value increases with increasing at a speed , it can be proved by the same token as that of Theorem 1 that (47) which implies that the estimate is consistent. This completes the proof. Proof of Lemma 3: When (18) is satisfied, it is obvious that (48)
(39) This is equivalent to
(49)
(40) Multiplying both sides of the aforementioned equation from the left by , and noting that , we have that
On the contrary, when have from the definitions of
and
This implies that there exists a stable TFM
As both and are outer according to their definitions, and are right coprime these relations imply that and can be internally stabilized by . Moreover, direct algebraic manipulations show that
(41)
(50)
is satisfied, we that
and are also normalized. Therefore, and are NRCFs On the contrary, assume that of a plant internally stabilizable by and satisfy . Then, both and are stable, and
(42)
(51)
such that
(52) (43)
is outer from its assumptions, the previous equation As satisfying further implies the existence of a stable TFM
(44)
as Define from (51), it can be easily proved that
. Then, (53)
Substituting this relation into (52), it can be further shown that (54)
ZHOU: FREQUENCY RESPONSE ESTIMATION FOR NCFS
This implies the existence of a TFM and
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satisfying
(55) Note that have
. We therefore
(56) From the previous relation and (53), it can be simply proved that
(57) This completes the proof. ACKNOWLEDGMENT The author would like to thank the associate editor and the reviewers for their constructive comments.
[7] W. P. Heath, “Probability density function of indirect nonparametric transfer function estimates for plants in closed-loop,” in Proc. 2000 SYSID, Santa Barbara, CA, 2000, pp. 321–326. [8] L. Ljung, System Identification: Theory for the User, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1999. [9] D. McFarlane and K. Glover, Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, ser. Lecture Notes in Control and Information Sciences. New York: Springer-Verlag, 1990, vol. 38. [10] P. Pintelon and G. Schoukens, System Identification: A Frequency Domain Approach. New York: IEEE Press, 2000. [11] M. B. Priestley, Spectral Analysis and Time Series. New York: Academic, 2001. [12] P. M. J. Van den Hof, R. J. P. Schrama, R. A. Callafon, and O. K. Bosgra, “Identification of normalized coprime factors from closed-loop experimental data,” Eur. J. Control, vol. 1, no. 1, pp. 62–74, 1995. [13] M. Vidyasagar, Control System Synthesis: A Factorization Approach. Cambridge, MA: MIT Press, 1985. [14] G. Vinnicombe, “Frequency domain uncertainty and the graph topology,” IEEE Trans. Autom. Control, vol. 38, no. 9, pp. 1371–1383, Sep. 1993. [15] J. S. Welsh and G. C. Goodwin, “Finite sample properties of indirect nonparametric closed-loop identification,” IEEE Trans. Autom. Control, vol. 47, no. 8, pp. 1277–1292, Aug. 2002. [16] T. Zhou, “On the computation and statistical properties of the MLE for the frequency response of NCFs,” Int. J. Control, vol. 76, no. 15, pp. 1547–1559, 2003. , “Nonparametric estimation for normalized coprime factors of a [17] MIMO system,” Automatica, vol. 41, no. 4, pp. 655–662, 2005. [18] T. Zhou and H. W. Xing, “Identification of normalized coprime factors through constrained curve fitting,” Automatica, vol. 40, no. 9, pp. 1591–1601, 2004.
REFERENCES [1] J. A. Ball, I. Gohberg, and L. Rodman, Interpolation of Rational Matrix Functions. Operator Theory: Advances and Applications. Basel, Germany: Birkhäuser, 1990, vol. 45. [2] D. R. Brillinger, Time Series: Data Analysis and Theory. San Francisco, CA: Holden-Day, 1981. [3] P. Data and G. Vinnicombe, “An algorithm for identification in the v-gap metric,” in Proc. 38th Conf. Decision and Control, Phoenix, AZ, 1999, pp. 3230–3235. [4] U. Forssell and L. Ljung, “Closed-loop identification revisted,” Automatica, vol. 35, no. 7, pp. 1215–1241, 1999. [5] T. Georgiou and M. C. Smith, “Optimal robustness in the gap metric,” IEEE Trans. Autom. Control, vol. 35, no. 5, pp. 673–686, May 1990. [6] G. X. Gu, “Modeling of normalized coprime factors with v -metric uncertainty,” IEEE Trans. Autom. Control, vol. 40, no. 8, pp. 1498–1511, Aug. 1999.
Tong Zhou (M’02) was born in Hunan Province, China, in 1964. He received the B.S. and M.S. degrees from the University of Electronic Science and Technology of China, Chengdu, China, in 1984 and 1989, respectively, and the Ph.D. degree from Osaka University, Osaka, Japan, in 1994. After visiting several universities in the Netherlands, China, and Japan, he joined Tsinghua University, Beijing, China, in 1999, where he is now a Professor of control theory and control engineering. His research interests include robust control, system identification, signal processing, hybrid systems, communication systems, and their applications to real-world problems. Dr. Zhou was a recipient of the first class natural science prize in 2003 from the Ministry of Education, China.
