ed on the basis of closed-loop data

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(1)ASYMPTOTIC VARIANCE EXPRESSIONS FOR CLOSED-LOOP IDENTIFICATION AND THEIR RELEVANCE IN IDENTIFICATION FOR CONTROL Michel Gevers z 1, Lennart Ljung x and Paul M.J. Van den Hof 2 ]. z CESAME, B^atiment Euler, Louvain University, B-1348 Louvain-la-Neuve,. Belgium. , x Department of Electrical Engineering, Link oping University, S-581 83 Link oping, Sweden. Mechanical Engineering Systems and Control Group, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands. ]. Abstract. Asymptotic variance expressions are analysed for models that are identi-. ed on the basis of closed-loop data. The considered methods comprise the classical 'direct' and 'indirect' method, as well as the more recently developed indirect methods, employing coprime factorized models and model parametrizations based on the dual Youla/Kucera parametrization. The variance expressions are compared with the open-loop situation, and evaluated in terms of their relevance for subsequent modelbased control design. Additionally it is specied what is the optimal experimental situation in identication (open-loop or closed-loop), in view of the variance of the resulting model-based controller.. Keywords. System identication closed-loop identication asymptotic variance expressions, prediction error methods model-based control design. 1. INTRODUCTION When identifying dynamic models for the specic purpose of subsequent model-based control design it is argued that a closed-loop experimental setup during the identication experiments supports the construction of an identied model that is particularly accurate in that frequency region that is relevant for the control design. This mechanism which plays a major role in many contributions in the area of \identication for control", has been motivated mainly on the basis of bias considerations in the form of a \control-relevant" distribution of the bias over frequency (Gevers, 1993 Van den Hof and M. Gevers acknowledges the nancial support of the Belgian Programme on Interuniversitary Poles of Attraction. 2 Author to whom correspondence should be addressed. E-mail: p.m.j.vandenhof@wbmt.tudelft.nl.. 1. Schrama, 1995). Recently it has been shown in Hjalmarsson et al. (1996), that for a particular class of control design methods, also from a variance point of view closed-loop experiments are preferred over open-loop ones. In this paper we will rst present the asymptotic variance expressions for identied models based on several dierent closed-loop identication methods, including the recently introduced indirect methods using a coprime factor model representation (Van den Hof et al., 1995) and the method employing a so-called dual Youla/ Kucera parametrization (Hansen and Franklin, 1988). The results for the classical 'direct' method (Ljung, 1993) are extended to also include variance expressions for the estimated noise model, while they are shown to remain the same for the mentioned alternative indirect methods. Consequences are shown for the variance of resulting model-based controllers for several types of controller designs..

(2) M, and both plant model and noise are estimated. In. 2. PRELIMINARIES We will consider the closed-loop conguration as depicted in Fig. 1, where G0 and C are linear time-invariant, possibly unstable, nite dimensional systems, with G0 strictly proper, while C is a stabilizing controller for G0 e is a white noise process with variance 0 , and H0 a stable and stably invertible monic transfer function. Signals r1 and r2 are external reference signals that are possibly available from measurements. For purpose of ecient notation, we will often deal with the signal r(t) := r1 (t) + C (q )r2 (t) being the result of external excitation through either r1 or r2 . Additionally we will denote: u(t) = u (t) + u (t) r. e. e. ?. r2. -d ;6. +. H0. r1. -? -. + + d. C. u. G0. v y. -? -. + + d. this case (Ljung, 1987): ^ n ;1 G(e ) ( ! ) (! ) cov ^ N (!) (!) 0 : H (e ) i!. u. v. i!. eu. ue. With the relation = ;CS0 H0 0 and using the fact that 0 ; j j2 = 0 it follows that ue. u. cov. r u. ue. ^ G H^. . n v N ru. 2 4. 3. (CS0 H0 ) 5 : (3) CS0 H0 1. u. 0. The variance expressions for G^ and H^ then become: n n ] cov (G^ ) = 1 + (4) N N ]: n = Nn 1 + (5) cov (H^ ) N v. v. e u. r u. u. r u. v. u. v. e u. 0. r u. 0. r u. The case of an open-loop experimental situation now appears as a special situation in which = 0, = , and C = 0, leading to the well known expressions n n cov (G^ ) cov (H^ ) : N N 0 e u. u. v. Fig. 1. Closed-loop conguration. with u (t) := S0 (q )r(t) u (t) := ;C (q )S0 (q )H0 (q )e(t) where the sensitivity function S0 is given by S0 (q) := 1 + C (q)G0 (q)];1 . For the corresponding spectra it follows that = + with = jS0 j2 and = jCS0 j2 : (1) r. e. r u. u. r u. e u. e u. r. v. The arguments q and e will be omitted when appropriate. We will consider parametrized models G(q ) for G0 and H (q ) for H0 with 2 , and we will use expressions S 2 M and G0 2 G to indicate the situations that both G0 and H0 or only G0 can be modelled exactly within the model set. The variance expressions that are considered in this paper are asymptotic in both n (model order) and N (number of data), while n=N is supposed to tend to 0, as in the standard framework of Ljung (1987). i!. 3. DIRECT IDENTIFICATION The direct method of closed-loop P identication is characterized by ^ = arg min 1 =0;1 "(t )2 with N. N. N. t. "(t ) = H (q );1 y (t) ; G(q )u(t)]:. r u. (2). An expression for the asymptotic variance of the transfer function estimate can be given for the situation that S 2. v. u. As indicated in Ljung (1993), the closed-loop expressions show that only the noisefree part u of the input signal contributes to variance reduction of the estimates. The given expressions are restricted to the situation that S 2 M and that both G() and H () are identied. r. Remark 3.1. The situation of estimating a plant model in the situation G0 2 G and having a xed and correct noise model H = H0 is considered in Ljung (1993) and is shown to be given by cov(G^ ) uv . This is a smaller variance than the situation in which both G and H are estimated. n. N. 4. INDIRECT IDENTIFICATION 4.1 Introduction Recently several dierent indirect approaches to closedloop identication have been presented, see e.g. Gevers (1993) and Van den Hof and Schrama (1995). These methods have been introduced from considerations related to the bias that occurs in closed-loop identication of approximate models. Here we will brie y illustrate their properties with respect to the variance of the estimates. 4.2 Coprime factor identication Coprime factor identication is treated in detail in Van den Hof et al. (1995). It is a scheme that relates to (and.

(3) generalizes) the classical joint input/output method of closed-loop identication as e.g. described in Gustavsson et al. (1977). It does not require knowledge of the implemented controller C . The basic principle is that the (two-times-two) transfer function (r e) ! (y u) is identied, while the plant ^ H^ ) are retrieved from these closed-loop estimodels (G mates. Consider the system's relations, using a ltered signal x(t) := F (q)r(t): T. T. y (t) = N0 F x(t) + S0 H0 e(t) u(t) = D0 F x(t) ; CS0 H0 e(t). := G0 S0 F ;1 and D0. (6) (7). := S0 F ;1 , consti-. with N0 tuting a coprime factor representation of G0 as G0 = N0 D0;1 . The linear and stable lter F can be chosen by the user to serve several purposes, like minimal order properties or normalization of the coprime factorization as discussed in Van den Hof et al. (1995) this will not be pursued here any further as it is immaterial for the variance analysis. The important observation here is that the signals x and e are uncorrelated. Identication of the 4 transfer functions in (6),(7) from the signals x(t), y(t), u(t) therefore corresponds to a one-input two-output open-loop identication problem. Denote F. F. F. F. "y (t ) = Wy (q );1 y (t) ; N (q )x(t)] "u (t ) = Wu (q );1 y (t) ; D(q )x(t)]. Least squares minimization of (" " ) provides esti^ D ^ W^ W^ . mated models N Open-loop models G^ and H^ are then retrieved by y. y. u. T. u. ^ );1 G^ = N^ (D ^ y: H^ = (1 + C G^ )W For the variance of G^ and H^ , use can be made of rst order approximations: G^ = G0 + !G, N^ = N0 + !N , ^ = D0 + !D etcetera, leading to D F. F. !G = D!N ; N0D2!D 0 0 !H = (1 + CG0 )!W + C (!G)W : F. F. (8). F. y. y. This leads to the result:. ^ n 2 1 (CS0 H0 ) 3 G cov ^ 4 CS0 H0 5 : (9) N H v. r u. u. 0. A sketch of the derivation of this result is given in the Appendix. Note that (9) is identical to expression (3) for direct identication .. 4.3 Identication in a dual Youla-Kucera parametrization Another method that has recently been introduced utilizes a specic parametrization of the plant G0 . As C stabilizes the plant, G0 can be parametrized within the class of all plants that are stabilized by C . This parametrization involves the relation N + D R ( ) (10) G() = D ; N R ( ) x. c. x. c. where N =D =: G is any (auxiliary) system that is stabilized by C N =D = C , and R() ranges over the class of all stable proper transfer functions. The dierent factors that build up the quotient expressions G and C are required to be stable and coprime. Using an expression like (10) for the plant G0 with a Youla-Kucera parameter R0 , and substituting this in the system's relations, shows -after some manipulationsthat these can be rewritten as z (t) = R0 x(t) + W0 e(t) x. x. x. c. c. x. with R0 = D S0 (G0 ; G )=D , W0 = H0 S0 =D , and x. x. c. c. z = (Dc + Gx Nc );1 (y ; Gx u) x = (Dx + CNx );1 r:. Since x is not correlated with e, the identication of R0 and W0 can again be considered as an open-loop identication problem. The signals z and x can be constructed by the user, as they are dependent on known quantities and measured signals. Least-squares identication is performed on the basis of the prediction error " (t ) = W (q );1 z (t) ; R(q )x(t)] z. and the estimated transfers are denoted by W^ and R^ . The open-loop model can then be reconstructed from these estimates according to N + D R^ (11) G^ = D ; N R^ ^ D S^;1 = W^ D 1 + C G^ ]: H^ = W (12) In order to guarantee that H^ is monic it will assumed that D is monic. Variance expressions for R^ and W^ are available through the standard (open-loop) expressions: x. c. x. c. c. c. c. cov (R^ ) . n jW0 j2 0 ^ ) n jW0 j2 and cov (W N x N. ^ W^ ) = 0. In a similar way as in section 4.2, while cov(R ^ H^ ) can be obtained, relying on rst the variance of (G order approximating expressions. Not surprisingly (see Appendix) the resulting expressions are again given by (9)..

(4) Further details on this identication method can be found in and Van den Hof and Schrama (1995). It can be shown that it is a direct generalization of the classical indirect method of closed-loop identication, see Van den Hof and De Callafon (1996). 4.4 Two-stage method A two-stage method for closed-loop identication has been introduced in Van den Hof and Schrama (1993). It operates directly on reference, input and output data, and does not require knowledge of the implemented controller. It can best be explained by considering the system's relations: y (t) = G0 ur (t) + S0 H0 e(t) u(t) = S0 r(t) ; CS0 H0 e(t):. In the rst step, measured signals r and u are used to estimate a model S^ of the sensitivity function S0 . Next this model is used to construct (by simulation) an estimate u^ of u according to u^ (t) = S^(q)r(t). In the second stage, the signals u^ and y are used as a basis for the identication of a plant model G^ . The procedure is very much alike the coprime factor identication scheme, albeit that the nal plant model is not calculated through division of two identied models this division is circumvented by constructing the auxiliary simulated signal u^ = S (q ^)r. Consider the prediction errors r. r. r. r. r. "y (t ) = Wy;1 y (t) ; G(q )S (q )r(t)] "u (t ) = Wu;1 u(t) ; S (q )r(t)]. then the parameter estimate ^ of this method can be written as the minimizing argument of V () for ! 1, with 1 X 1 "2 (t ) + "2 (t )] V () = N. N. N. N. N. =1. t. . y. u. (Note that for ! 1, ^ will be determined fully on the basis of r and u). Applying the coprime factor results from section 4.2 to this situation then shows that the variance results are equivalent, and independent of . 3 4.5 Summarizing comments For the considered indirect methods, the asymptotic variance expressions for plant and noise model are exactly the same as the expressions for direct identication. This may not be too surprising, as similar results The authors acknowledge the contribution of Urban Forssell (Univ. Linkoping) to the proof of this result. 3. for the classical indirect and joint i/o methods were already available (Gustavsson et al., 1977). However what has to be stressed here, is that for the indirect type methods the variance expressions for G^ are valid also in the situation that G0 2 G but S 2= M, while for the direct method the results are only achieved under the stronger condition that S 2 M. 5. OPEN-LOOP VERSUS CLOSED-LOOP EXPERIMENTS Considering that the variance expressions are identical for all closed-loop identication methods, we can now make a comparison between the variances obtained from open-loop and closed-loop experimental conditions. The appropriate expressions are summarized in table 1. Open-loop n v N u n v. Closed-loop. n v < N. ru e < Nn v 1 +. ur N 0 0 u Table 1. Variance expressions under openloop and closed-loop conditions. V ar(G^ N ) V ar(H^ N ). The results show that for both G^ and H^ the variance obtained under closed-loop identication will generally be larger than for open-loop identication. Particularly in a situation where the input power is limited, the difference will become apparent, as in that case only part of the actual input spectrum can be used for variance reduction of G^ and H^ . In case the input power is not restricted, closed-loop identication can achieve the same results as open-loop identication, by choosing a reference signal r such that is equal to the input spectrum applied in the open-loop situation. The results suggest that in terms of variance of the model estimates G^ and H^ , open-loop identication always has to be preferred over closed-loop identication. However, perhaps surprisingly, this is not the case if the objective of the identication is model-based control design, as is explained in the next section. r u. N. N. 6. OPTIMAL EXPERIMENTS IN VIEW OF MODEL-BASED CONTROL In this section we will consider the situation that the identied transfer functions G^ and H^ are used as a basis for model-based control design, and we will illustrate the eect of the variance of the identied model on the model application, i.e. the designed controller. To this end we will rst consider the following result from Ljung (1987, Theorem 14.3). N. N.

(5) Proposition 6.1. Consider the variance-based identication design criterion J (D) =. Z. . ;. N. trP (! D);(! )]d!. . where P (! D) = covG^ (e ) H^ (e )] , D denotes the design choices with respect to the experimental conditions, represented by f g, while ;(!) is a 2 2 Hermitian matrix re ecting the intended application of the model. If ;12 (!) 0 and the input power is limited, then the experimental condition D for which J (D) is optimized is given by p = c ;11 (!) (!) 0 i!. u. i!. T. ue. opt u. opt ue. v. 2. and c is a constant.. This result shows that open-loop identication is optimal when in the intended model application, the covariance between G^ and H^ is not penalized, but only the variance contributions of G^ and H^ separately. This situation applies e.g. to the case where a controller is designed on the basis of G^ only. Corollary 6.2. Consider as model application a control design scheme based on a frequency weighted sensitivity minimization: C ^ = arg min kV (1 + C~ G^ );1 k2 : G. ~. C. Then the optimal experiment design in line with the above proposition is given by open-loop experiments p ( 0). 2 = c jC ^ V S0 j opt ue. opt u. G. v. Proof. The application-related error criterion can be written as kV (1 + CG0 );1 ; (1 + C G^ );1 ]k2 which can be shown to be equal to (using rst order approximations) k (1+( 0 ;0 )^2) k2 . An appropriate choice of ;11 for this model 2 application would thus be ;11 (!) = j1+j j 0 j4 leading to the result presented. 2 From the above result one could conclude that -from a variance point of view- open-loop identication is optimal for this control design. However, the required input spectrum in this `open-loop' situation should be proportional to the sensitivity function S0 of the real plant, being controlled by the yet-to-be-designed controller. Input shaping with S0 is exactly what is done in closedloop identication, since = jS0 j2 + . A second related result is present in the recent work of Hjalmarsson et al.(1996) on optimal identication for VC G. G. CG. VC. CG. u. r. e u. control. In this work the identication criterion is selected to minimize the control performance degradation that results from the random errors on G^ and H^ . In solving this problem, the authors have quantied the variance error on the designed model-based controller. Consider a situation where an identied model G^ , H^ is obtained from a closed-loop experimental situation with a controller C implemented on the plant. Consider a model-based control design scheme C^ = c(G^ H^ ) N. N. N. id. N. N. N. and let F , F re ect the derivatives of c with respect to G, H , i.e. the sensitivity of the controller with respect to changes in G and H . Then the variance of the controller estimate is (see Hjalmarsson et al., 1996) G. H. n jH0 j2 N

(6). jFH j2 + 0 jFG + (FG G0 + FH H0 )Cid j2. cov (C^N ) . leading to the following situations. Situation F 6= 0. The controller variance is minimized for models identied in closed-loop with an implemented controller C unequal to zero, and the resulting controller variance is n cov (C^ ) jH0 j2 jF j2 : N r. H. opt. id. N. H. By comparison, the controller variance obtained with open-loop identication is. . . n jF j2 0 : cov (C^ ) jH0 j2 jF j2 1 + N jF j2 N. G. H. H. u. We observe that the variance obtained under ideal closedloop experimental conditions can only be achieved with open-loop identication if the input power is made innite. Situation F = 0. The variance expression becomes j1 + C G0 j2 0 = n jF j2 : n cov (C^ ) jH0 j2 jF j2 N N H. id. v. G. N. r u. r. G. The corresponding expression for open-loop identication is n jF j2 : cov (C^ ) N v. G. N. u. The situation F = 0 means that the control design depends only on G and not on the noise model. This result is therefore consistent with Corollary 6.2. We conclude from this analysis that, as far as variance errors are concerned, for model-based control design, closed-loop identication is optimal except when the controller is independent of the noise model. H.

(7) 7. CONCLUSIONS Asymptotic variance expressions have been derived for several closed-loop identication schemes, involving both the (classical) direct method and more recently introduced indirect identication methods. It is shown that the several approaches lead to the same asymptotic variance. Although asymptotic variance of plant model and noise model generally will increase when performing closed-loop identication, in comparison with open-loop identication, closed-loop identication can still be preferred when the identied model is used as a basis for control design. In the case that a controller is designed on the basis of both plant model and noise model, closedloop identication is shown to lead to better variance results.. c. x. and so. c. c. c. . cov (G^ ) = . x. Dc. Dx S02 (1 + CGx. x. 2 ^ ) cov(R):. Substituting the expression for cov(R^ ) and using the property that = jD (1 + CG )j2 it follows after some manipulation that cov(G^ ) n=N = . For cov(H^ ) it follows from (A.4) that jD j2 covW^ + jN W j2 covG: ^ cov (H^ ) = 0 jS j2 x. x. x. r. v. r u. c. 0. c. Substituting the known expressions in the right hand side, will show that cov(H^ ) n=N jH0 j2 1 + = ]. ^ H^ ) it follows from (A.4) that cov(G ^ H^ ) = For cov(G (W0 N ) cov(G^ ) which leads to the appropriate result. e u. APPENDIX. Proof of (9).. r u. c. Applying the standard variance expressions to the multivariable situation of (6),(7) it follows that. ^. n jS0 j2 v N ^ N x D ^ n jS0 j2 v W cov ^ y N 0 Wu cov. For !G this leads to D (!R) D + G0 N !G = D ; N R0 !R = D S02 (1 + CG ). . 1 ;C . . (A.1) ;C jC j2 1 ;C : (A.2) ;C jC j2. Since (6),(7) re ect an open-loop situation (as x and e are uncorrelated) this implies that the cross-covariance ^ D^ ) and (W^ W^ ) are zero. From the terms between (N rst order approximations in (8) it follows that j!Gj2 = T. y. u. j!N j2 + jG0 j2 j!Dj2 ; 2Re

(8) G0 (!D)(!N ) : jD0 j2 jD0 j2 jD0 j2 F. F. F. Substitution of (A.1) then provides the result for cov(G^ ). For H^ one can similarly write (when neglecting terms that have expectation 0): j!H j2 = j1 + CG0 j2 j!W j2 + jCW j2 j!Gj2 (A.3) y. y. and the result for cov(H^ ) follows after substitution of ^ H^ ) follows from (A.2). The expression for cov(G ^ H^ ) = ;(CW ) cov(G^ ). cov (G y. Variance result for dual Youla-Kucera method. Using (11),(12) the related expressions for the rst order approximation errors become R) + (N + D R0 )N (!R) !G = (D ; N R0 )D ((! D ; N R0 )2 !H = D (!W ) + W N (!G): (A.4) x. c. c. x. x. c. S0. c. 0. c. c. c. REFERENCES Gevers, M. (1993). Towards a joint design of identication and control? In: H.L. Trentelman and J.C. Willems (Eds.), Essays on Control: Perspectives in the Theory and its Applications. Birkh&auser, Boston, pp. 111-151. Gustavsson I., L. Ljung and T. S&oderstr&om (1977). Identication of processes in closed loop - identiability and accuracy aspects. Automatica, 13, 59-75. Hansen, F.R. and G.F. Franklin (1988). On a fractional representation approach to closed-loop experiment design. Proc. American Control Conf., Atlanta, GA, USA, pp. 1319-1320. Hjalmarsson, H., M. Gevers and F. De Bruyne (1996). For model-based control design, closed-loop identication gives better performance. Automatica, 32, 16591673. Ljung, L. (1987). System Identication: Theory for the User. Prentice-Hall, Englewood Clis, NJ. Ljung, L. (1993). Information contents in identication data from closed-loop operation. Proc. 32nd IEEE Conf. Dec. Contr., San Antonio, TX, pp. 2248-2252. Van den Hof, P.M.J., R.J.P. Schrama, R.A. de Callafon and O.H. Bosgra (1995). Identication of normalised coprime plant factors from closed-loop experimental data. Europ. J. Control, 1, 62-74. Van den Hof, P.M.J. and R.J.P. Schrama (1995). Identication for control - closed-loop issues. Automatica, 31, 1751-1770. Van den Hof, P.M.J. and R.A. de Callafon (1996). Multivariable closed-loop identication: from indirect identication to dual-Youla parametrization. Proc. 35th IEEE Conf. Dec. Contr., Kobe, Japan, pp. 1397-1402..

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