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Closed-loop Identication Revisited { Updated Version Urban Forssell and Lennart Ljung Department of Electrical Engineering Linkping University, S-581 83 Linkping, Sweden WWW: http://www.control.isy.liu.se Email: ufo@isy.liu.se, ljung@isy.liu.se April 1998 REGL AU ERTEKNIK OL TOM ATIC CONTR LINKÖPING Report no.: LiTH-ISY-R-2021 Submitted to Automatica Technical reports from the Automatic Control group in Linkping are available by anonymous ftp at the address ftp.control.isy.liu.se This report is contained in the compressed postscript le 2021.ps.Z Closed-loop Identication Revisited { Updated Version ? Urban Forssell and Lennart Ljung Division of Automatic Control, Department of Electrical Engineering, Linkoping University, S-581 83 Linkoping, Sweden URL: http://www.control.isy.liu.se/ Abstract Identication of systems operating in closed loop has long been of prime interest in industrial applications The problem oers many possibilities, and also some fallacies, and a wide variety of approaches have been suggested, many quite recently The purpose of the current contribution is to place most of these approaches in a coherent framework, thereby showing their connections and display similarities and dierences in the asymptotic properties of the resulting estimates The common framework is created by the basic prediction error method, and it is shown that most of the common methods correspond to dierent parameterizations of the dynamics and noise models The so called indirect methods, e.g., are indeed "direct" methods employing noise models that contain the regulator The asymptotic properties of the estimates then follow from the general theory and take dierent forms as they are translated to the particular parameterizations In the course of the analysis we also suggest a projection approach to closed-loop identication with the advantage of allowing approximation of the open loop dynamics in a given, and user-chosen frequency domain norm, even in the case of an unknown, non-linear regulator Key words: System identication Closed-loop identication Prediction error methods This paper was not presented at any IFAC meeting Corresponding author U Forssell Tel +46-13-282226 Fax +46-13-282622 E-mail ufo@isy.liu.se ? Preprint submitted to Elsevier Preprint April 1998 Noise - Extra input fInput ++ - -? ++ f Plant Controller - Output ?f- + Set point Fig A closed-loop system Introduction 1.1 Motivation and Previous Work System identication is a well established eld with a number of approaches, that can broadly be classied into the prediction error family, e.g,, 22], the subspace approaches, e.g., 31], and the non-parametric correlation and spectral analysis methods, e.g., 5] Of special interest is the situation when the data to be used has been collected under closed-loop operation, as in Fig The fundamental problem with closed-loop data is the correlation between the unmeasurable noise and the input It is clear that whenever the feedback controller is not identically zero, the input and the noise will be correlated This is the reason why several methods that work in open loop fail when applied to closed-loop data This is for example true for the subspace approach and the non-parametric methods, unless special measures are taken Despite these problems, performing identication experiments under output feedback (i.e in closed loop) may be necessary due to safety or economic reasons, or if the system contains inherent feedback mechanisms Closed-loop experiments may also be advantageous in certain situations: In 13] the problem of optimal experiment design is studied It is shown that if the model is to be used for minimum variance control design the identication experiment should be performed in closed-loop with the optimal minimum variance controller in the loop In general it can be seen that optimal experiment design with variance constraints on the output leads to closed-loop solutions In \identication for control" the objective is to achieve a model that is suited for robust control design (see, e.g., 7, 19, 33]) Thus one has to tailor the experiment and preprocessing of data so that the model is reliable in regions where the design process does not tolerate signicant uncertainties The use of closed-loop experiments has been a prominent feature in these approaches Historically, there has been a substantial interest in both special identication techniques for closed-loop data, and for analysis of existing methods when applied to such data One of the earliest results was given by Akaike 1] who analyzed the eect of feedback loops in the system on correlation and spectral analysis In the seventies there was a very active interest in questions concerning closed-loop identication, as summarized in the survey paper 15] See also 3] Up to this point much of the attention had been directed towards identiability and accuracy problems With the increasing interest "identication for control", the focus has shifted to the ability to shape the bias distribution so that control-relevant model approximations of the system are obtained The surveys 12] and 29] cover most of the results along this line of research 1.2 Scope and Outline It is the purpose of the present paper to \revisit" the area of closed-loop identication, to put some of the new results and methods into perspective, and to give a status report of what can be done and what cannot In the course of this expose, some new results will also be generated We will exclusively deal with methods derived in the prediction error framework and most of the results will be given for the multi-input multi-output (MIMO) case The leading idea in the paper will be to provide a unied framework for many closed-loop methods by treating them as dierent parameterizations of the prediction error method: There is only one method The dierent approaches are obtained by dierent parameterizations of the dynamics and noise models Despite this we will often use the terminology \method" to distinguish between the dierent approaches and parameterizations This has also been standard in the literature The organization of the paper is as follows Next, in Section we characterize the kinds of assumptions that can be made about the nature of the feedback This leads to a classication of closed-loop identication methods into, so called, direct, indirect, and joint input-output methods As we will show, these approaches can be viewed as variants of the prediction error method with the models parameterized in dierent ways A consequence of this is that we may use all results for the statistical properties of the prediction error estimates known from the literature In Section the assumptions we will make regarding the data generating mechanism are formalized This section also introduces the some of the notation that will be used in the paper Section contains a brief review of the standard prediction error method as well as the basic statements on the asymptotic statistical properties of this method: Convergence and bias distribution of the limit transfer function estimate Asymptotic variance of the transfer function estimates (as the model orders increase) Asymptotic variance and distribution of the parameter estimates The application of these basic results to the direct, indirect, and joint inputoutput approaches will be presented in some detail in Sections 5{8 All proofs will be given in the Appendix The paper ends with a summarizing discussion in Section Approaches to Closed-loop Identication 2.1 A Classication of Approaches In the literature several dierent types of closed-loop identication methods have been suggested In general one may distinguish between methods that (a) Assume no knowledge about the nature of the feedback mechanism, and not use the reference signal r(t) even if known (b) Assume the feedback to be known and typically of the form u(t) = r(t) ; K (q)y(t) (1) where u(t) is the input, y(t) the output, r(t) an external reference signal, and K (q) a linear time-invariant regulator The symbol q denotes the usual shift operator, q;1y(t) = y(t ; 1), etc (c) Assume the regulator to be unknown, but of a certain structure (like (1)) If the regulator indeed has the form (1), there is no major dierence between (a), (b) and (c): The noise-free relation (1) can be exactly determined based on a fairly short data record, and then r(t) carries no further information about the system, if u(t) is measured The problem in industrial practice is rather that no regulator has this simple, linear form: Various delimiters, anti-windup functions and other non-linearities will have the input deviate from (1), even if the regulator parameters (e.g PID-coecients) are known This strongly disfavors the second approach In this paper we will use a classication of the dierent methods that is similar to the one in 15] See also 26] The basis for the classication is the dierent kinds of possible assumptions on the feedback listed above The closed-loop identication methods correspondingly fall into the following main groups: (1) The Direct Approach : Ignore the feedback and identify the open-loop system using measurements of the input u(t) and the output y(t) (2) The Indirect Approach : Identify some closed-loop transfer function and determine the open-loop parameters using the knowledge of the controller (3) The Joint Input-Output Approach : Regard the input u(t) and the output y(t) jointly as the output from a system driven by the reference signal r(t) and noise Use some method to determine the open-loop parameters from an estimate of this system These categories are basically the same as those in 15], the only dierence is that in the joint input-output approach we allow the joint system to have a measurable input r(t) in addition to the unmeasurable noise e(t) For the indirect approach it can be noted that most methods studied in the literature assume a linear regulator but the same ideas can also be applied if non-linear and/or time-varying controllers are used The price is, of course, that the estimation problems then become much more involved In the closed-loop identication literature it has been common to classify the methods primarily based on how the nal estimates are computed (e.g directly or indirectly using multi-step estimation schemes), and then the main groupings have been into \direct" and \indirect" methods This should not, however, be confused with the classication (1)-(3) which is based on the assumptions made on the feedback Technical Assumptions and Notation The basis of all identication is the data set Z N = fu(1) y(1) : : : u(N ) y(N )g (2) consisting of measured input-output signals u(t) and y(t), t = 1 : : : N We will make the following assumptions regarding how this data set was generated Assumption The true system S is linear with p outputs and m inputs and given by y(t) = G0(q)u(t) + v(t) v(t) = H0(q)e(t) (3) where fe(t)g (p 1) is a zero-mean white noise process with covariance matrix 0, and bounded moments of order 4+ , some > 0, and H0(q) is an inversely stable, monic lter For some of the analytic treatment we shall assume that the input fu(t)g is generated as u(t) = r(t) ; K (q)y(t) (4) where K (q) is a linear regulator of appropriate dimensions and where the reference signal fr(t)g is independent of fv(t)g This assumption of a linear feedback law is rather restrictive and in general we shall only assume that the input u(t) satises the following milder condition (cf 20], condition S3): Assumption The input u(t) is given by u(t) = k(t yt ut;1 r(t)) (5) where y t = y (1) : : : y(t)], etc., and where where the reference signal fr(t)g is a given quasi-stationary signal, independent of fv(t)g and k is a given deterministic function such that the closed-loop system (3) and (5) is exponentially stable, which we dene as follows: For each t s t s there exist random variables y s(t) u s(t), independent of rs and v s but not independent of rt and vt, such that E ky(t) ; y s(t)k4 < Ct;s E ku(t) ; u s(t)k4 < Ct;s (6) (7) for some C < 1 < In addition, k is such that either G0 (q) or k contains a delay Here we have used the notation N X Ef (t) = Nlim Ef (t) !1 N t=1 (8) The concept of quasi-stationarity is dened in, e.g., 22] If the feedback is indeed linear and given by (4) then Assumption means that the closed-loop system is asymptotically stable Let us now introduce some further notation for the linear feedback case By combining the equations (3) and (4) we have that the closed-loop system is y(t) = S0(q)G0 (q)r(t) + S0 (q)v(t) (9) where S0(q) is the sensitivity function, S0(q) = (I + G0(q)K (q));1 This is also called the output sensitivity function With Gc0(q) = S0(q)G0(q) and Hc0(q) = S0 (q)H0(q) we can rewrite (9) as y(t) = Gc0(q)r(t) + vc(t) vc(t) = Hc0(q)e(t) In closed loop the input can be written as u(t) = S0i (q)r(t) ; S0i (q)K (q)v(t) = S0i (q)r(t) ; K (q)S0 (q)v(t) The input sensitivity function S0i (q) is dened as S0i (q) = (I + K (q)G0(q));1 (10) (11) (12) (13) (14) (15) The spectrum of the input is (cf (14)) u = S0i r (S0i ) + KS0 v S0K (16) where r is the spectrum of the reference signal and v = H0 0H0 the noise spectrum Superscript denotes complex conjugate transpose Here we have suppressed the arguments ! and ei! which also will be done in the sequel whenever there is no risk of confusion Similarly, we will also frequently suppress the arguments t and q for notational convenience We shall denote the two terms in (16) r u = S0i r (S0i ) (17) e u = KS0 v S0K = S0i K v K (S0i ) (18) and The cross spectrum between u and e is = ;KS0 H0 0 = ;S0i KH0 0 The cross spectrum between e and u will be denoted eu, (19) ue eu = ue Occasionally we shall also consider the case where the regulator is linear as in (4) but contains an unknown additive disturbance d: u(t) = r(t) ; K (q)y(t) + d(t) (20) The disturbance d could for instance be due to imperfect knowledge of the true regulator: Suppose that the true regulator is given by K true(q) = K (q) + !K (q) (21) for some (unknown) function !K In this case the signal d = ;!K y Let rd ( dr ) denote the cross spectrum between r and d (d and r), whenever it exists Prediction Error Identication In this section we shall review some basic results on prediction error methods, that will be used in the sequel See Appendix A and 22] for more details 4.1 The Method We will work with a model structure M of the form y(t) = G(q )u(t) + H (q )e(t) (22) G will be called the dynamics model and H the noise model We will assume that either G (and the true system G0) or the regulator k contains a delay and that H is monic The parameter vector ranges over a set DM which is assumed compact and connected The one-step-ahead predictor for the model structure (22) is 22] y^(tj) = H ;1(q )G(q )u(t) + (I ; H ;1(q ))y(t) (23) The prediction errors are "(t ) = y(t) ; y^(tj) = H ;1(q )(y(t) ; G(q )u(t)) (24) Given the model (23) and measured data Z N we determine the prediction error estimate through ^N = arg min V ( Z N ) (25) 2DM N N X N VN ( Z ) = N "TF (t ) ;1"F (t ) (26) t=1 "F (t ) = L(q )"(t ) (27) Here is a symmetric, positive denite weighting matrix and L a (possibly parameter-dependent) monic prelter that can be used to enhance certain frequency regions It is easy to see that "F (t ) = L(q )H ;1(q )(y(t) ; G(q )u(t)) (28) Thus the eect of the prelter L can be included in the noise model and L(q ) = can be assumed without loss of generality This will be done in the sequel We say that the true system is contained in the model set if, for some 0 DM , G(q 0) = G0(q) H (q 0) = H0 (q) (29) This will also be written S M The case when the true noise properties cannot be correctly described within the model set but where there exists a 0 DM such that G(q 0) = G0(q) (30) will be denoted G0 G 4.2 Convergence Dene the average criterion V () as T (t ) ;1"(t ) V () = E" (31) Then we have the following result (see, e.g., 20, 22]): ^N ! Dc = arg min V () with probability (w p.) as N ! (32) 2DM In case the input-output data can be described by (3) we have the following characterization of Dc (G is short for G(q ), etc.): 3 ( G ; G ) Dc = arg min tr ;1H;1 64 75 64 2DM ; (H0 ; H ) Z h u eu 32 ( G ; G ) ue ; i 54 H d! 0 (H0 ; H ) This is shown in Appendix A.1 Note that the result holds regardless of the nature of the regulator, as long as Assumptions and hold and the signals involved are quasistationary From (33) several conclusions regarding the consistency of the method can be drawn First of all, suppose that the parameterization of G and H is suciently #exible so that S M If this holds then the method will in general give 10 (33) ... System identication Closed- loop identication Prediction error methods This paper was not presented at any IFAC meeting Corresponding author U Forssell Tel +4 6-1 3-2 82226 Fax +4 6-1 3-2 82622 E-mail... April 1998 Noise - Extra input fInput ++ - -? ++ f Plant Controller - Output ?f - + Set point Fig A closed- loop system Introduction 1.1 Motivation and Previous Work System identication is... the open-loop system parameters from the closed- loop model obtained in step 1, using the knowledge of the regulator Instead of identifying the closed- loop system in the rst step one can identify