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Closed-Loop Identification Methods for LQ Control Design
Copyright ~ IFAC System Identification, Copenhagen, Denmark, 1994
CLOSED-LOOP IDENTIFICATION METHODS FOR LQ CONTROL DESIGN A.C. VAN DER KLAUW·,t, J.E.F. VAN OSCH· and P.P.J. VAN DEN BOSCH·· °Delft Uaiwm', 0/ Tecltaolon. De,.. 0/ Eleetric.l Ea,., Coa'rol Ldo,..'orr. P.D. Bft SO'l. liDO CA DelfI. Tle NetAerl..4.
.oEiaaOtlefl Uaitlerftt, 0/ Tecltao/on. De,.. 0/ Eleetric.l Ea,.• M_remea' MB Eiaao"ea. Tle NetAerl.ah
.a~
Coa'rol Sedioa. P.D. Bn Sl'. S600
Abatrae&. ID CODDedion with the current intereat in the interae\ion between identification and control deaip, a clc.ed-loop identification method is propoaed that take- into account a aubeequent LQ control deaip mp. It appears that tbia method unifiea emtin& clc.ed-loop identification acheme-. Hence it facililatea comparison of tbae acheme-. The propert.iea of the rsultinl model depend OD the chc.en parametriaation. It is ahown that the npt choice of parametrisation reau1ta in a model, of which the biu distribution is independent of the noi8e, which is an advant&«e with reapect to aome d_ca1 approachea. The method is illuatrated with a aimulation eumple.
Key Worda. Clc.ecl-loop ayItema, multivariable ayItema, identification for control, eaiimation theory, parameter MtimatioD. ~ identificatioD. .
loop excitation signals r, (reference signal) or e, (extemal input). The process output is 11" and the process input is u,. The dimensions of the signals are: r"11" W",: P x 1; e" u" W ..,': m x 1. The transfer function matrices have appropriate dimensions.
1. INTRODUCTION
The problem of identifying multivariable processes operating under feedback has attracted much attention off late (Bitmead et al., 1990; Geven, 1993; Van den Hof and Schrama, 1993; Zang et al., 1991). This has been inspired by the wish to make better use of the interaction between identification and control design. The key idea is that in the identification step the control step should already be taken into account. To improve a controller, it is argued by e.g. Gevers (1993) that the model should be obtained from closed-loop data. Another reason for doing identification under feedback is th!,-t unstable systems can be stabilized. For the process industry an additional reason is that output quality can be maintained during the experiment, thus saving costs.
We assume that the controller {C" C,} needs improvement, and that it is obtained by minimizing the LQ criterion JC(C"C,):
JC(C" C.) =
~ t {(r, -;,(Cb C.; 8»'(r, -;,(C" C.i 8» &=1
+AU,(Cb C.J)'u,(C" C.; 8)}
where' denotes the transpose of a matrix, and where A is a positive constant. Note that the controller {Cb C.} is calculated on the basis of a model G(8) of the real plant Go.
This paper is concerned with identifying a model of a plant, that is operating in closed loop. In Fig. 1 a typical closedloop situation is depicted. The plant is denoted by Go(q),
The problem we address in this paper is to obtain a model G(q,8) of the plant Go(q), from N measurements of 11" u" re and e" that is suitable for control design according to (1). The model is parametrized with parameter vector 8; and 8 is the estimated parameter vector. It is explicitly assumed that the process Go is not in the model set, which implies that we do not only want the identification procedure to give consistent estimates asymptotically, but we also need tunable expressions for the asymptotic bias distribution. Problems are caused by the correlation between the process input and the noise. In oven-loop identification this correlation is not present. However, from Fig. 1 it follows that this is not true in closed·loop situations.
Fil. 1. Typical cIoIed-loop COllfiIuration
where q-I is the well-known delay operator. For readability q is often omitted in the equations in this paper. The controller is a two-desree of freedom controller, denoted by CJ(q) and C.(q). All disturbances are lumped into the noise signals w", and w..,. It is supposed that the loop is excited by at least one of the • ne reoearda of A.C.
ftII
(1)
In the following section we will briefly trea.t some classical closed-loop identification procedures, that are not tailored for control design. We then come to the formulation of an identification criterion, suitable for LQ control design.
cIer IQuw ia made poMiblo by ~e DlIkh Tedl-
aaiocY FoaadUioa ucler Grut STW DEL92.1nll
1525
is obtained with a Prediction Error Method (PEM) as:
Subsequently, in section 3 we will show that the proposed method also includes several classical closed-loop identification schemes. The properties of the asymptotic models depend on the particular parametrization, which is illustrated by deriving the asymptotic bias distribution for different parametrizations. A simulation example illustrates the method, and we draw some conclusions in section 5.
[I Go] Wl
,
ti = arg min In(I) In(8) =
(2)
/(8)
=
C/~,(w)Ci
+C/~"(w)
(3)
Ul
(5)
Ut 1/t
where· denotes the conjugated transpOlle of a complex matrix, and with ~, the spectrum of rh ~. the spectrum of eh and ~ .. and ~ .. their cross-spectra. We now define the dOlled.loop traM/er junction go as:
L
' '} N1 N (I {A(I)Yl - B(8)ul
(lOa) (lOb)
=
Ul+V...t SO"'t GOUl +Vr,l
(Ba) (Bb) (Bc)
In the first step an estimate S of the sensitivity function So is calculated. With this estimate the noise-free signal Ul is calculated. Observing that in (Bc) the signal Ut and the noise Vr,l are uncorreiated, an estimate G can be obtained from open-loop calculations. Although this method asymptotically yields a consistent estimate, it is not clear how the first step influences the second step. A thorough analysis is not yet available. More about this method will be said in section 3.2.
(6)
so that finally we have Zl = .il + VI = YO"'1 + VI
=
= 0)
Recently a new method was proposed by Van den Hof and Schrama (1993), called the TIDO-Step method (TS). For SISO systems, an expreaion for the asymptotic bias distribution was derived, in the case where the system is not in the model set. The method is based on the following relations:
will be used later, and is given
+ ~.(w) + ~ .. (w)Ci
(9c) (9d)
=
(4)
~,,(w)
(9b)
tsl
with G(8) ..4-1 (I)B(I). From (8)-(10) it is clear that the classical methods are variatfons of the well-known open-loop methods. For closed-loop systems they have two disadvantages. First of all, if the process is not in the model set, the resulting model is biased, and the bias depends on the noise. Secondly, they do not take into account the purpose of the model: control design.
"'t
~,,(w),
(rl - YI(I/»'(rl - Yl(l/»
(9a)
1=1
In general the controller {C" C.} is known, and Cl is stable. Hence we can define a loop excitation signal as
The spectrum of"'h by
t
~1(J(8)
=
= [ ~o ] SO{Clrl +~} + VI
(Se)
The plant model G(ti) is obtained by solving (9d). Using IMtrumental Varia61e (IV) techniques, a model of the plant can be obtained as well (SOderatrom, 1987). However, the construction of the instruments is more involved than in the open-loop case, since the input Ul is correlated with the noise. Provided appropriAte instruments (I can be found, the parameter estimate ti is obtained as
=
Zl
~
il(8) = GS(8)"'1 GS(ti) = G(ti)[I + C.G(ti»)
Let us define z(t) (r: u:)'. The first part of (2) can be seen as the noise-free part of z(t), and will be denoted by .il . The second part of (2) is the noise contribution, denoted by VI (V~.l V~.l)'. Furthermore the loop aeMitit7ity junction So is defined as So [I + C.GO)-I. We then obtain the following:
=
(8b)
The disadvantage of this method is the dependence of ti on the noise. This will be shown in eection 3. With Indirect Identification (11) first a model of the cloeed loop is estimated, between the loop excitation signal "" and the output rl' Accurate knowledge of the controller then yields a model of the plant:
[Cl I] ( :: ) l
(YI - Yl(8»'(rl - Yl(I/»
iM) = G(8)uI
Let us first derive the equations that describe the loop behaviour in Fig. 1. Defining the noise vector Wl as Wt = (W~.l w~,t)', the equations can be written as:
[_~.] [I + GoC.r
~t
(Sa)
1=1
In this eection we will briefly treat some classical cloeedloop identification methods, that methods do not take into account that the model will be used for control desipl. Therefore an extension is proposed, that is clOllely related to LQ-type controllers. Within this method we can vary the parametrlzation, which influences the properties of the resulting model. This will become clear in the sequel.
+
= argminJOI(8)
JOI(8)
2. CLOSED-LOOP IDENTIFICATION
( ~:) = [~o] [I + C.Gor l
,
ti
(7)
Without taking into account control design, the closed-loop identification problem is the problem of obtaining a model G(ti) of Go from cloeed-loop d&ia. In recent years, eeveral procedures for cloeed-loop identification have been proposed (e.g. Gustavsson et al., 1977; SOderatrom et al. 1975, 1987). The mostly used method is probably Direct Identification (DJ). Measurements of both rl and Ul are taken, and procesaed without taking into ac: count the presence of feedback. The parameter estimate 8
To improve the identification results, in this paper we proa cloeed-loop identification method that takes into account both LQ control desipl and undermodelling. It appears to unify DJ, 11 and TS, where the two steps in the TS method are taken together, so that analysis is now p0ssible. The main ingredie!lt i~ the d~tion of the global pOlIe
1526
~ f, {(YI -li,)'(1I1 -li,)
J1(8) =
The criterion (12) then becomes
1:1
+~( tI, -
UI)'( tI,
-
UI)}
J1(8) =
(12)
t._.(w) = 0
~G(8) = Go - G(8) 10
=
~(8)Sot~{~G(8)So}O
J!.,(8)
1 -G(8)
It.. ! I
1°
(19)
-G(8)
1° dw
(20)
with (21) Minimization of J:"(8) will result in a small model error ~G(8), but the model quality is dependent on the noise, due to the 8-dependence of the noise contribution in (20). This effect is of course unwanted.
3.2. Separate Parametrization of' and
u
The second possible parametrization is
(13&)
u,(P) =
,,(8, P) =
S(P)T/,
G(8)ul(P)
(22&) (22b)
To derive J:"(8, P), we will calculate tI, - U, and ", - ',. We see from (6)-(7) and the parametrization (22) that
(13b)
tll - iI,(P)
tI, -
S(P)T/,
= {So - S(P)}"d v." Y, - ,,(8,P)
=
(23)
Y, - G(8)S(P)T/1 {GoSo - G(8)S(P)}T/' + v",
(24)
We define ~s(P) = So - S(P) ~(8) 10
=
(25b)
that we obtain J!.,(8,P)
3.1. Parametrization of' Only
(25&)
Go - G(8)
= tr [ Q,(8,P)t~Q;(8,P)
+.\Q.(P)t~Q:(P)
+ t .. + .\t.. dw
(26)
with
The first parametrization is tI,
-G(8)
= tr J~ Q,(8)t~Q;(8)
+!
where w.p. 1 means tDith probability one, tr is the trace of a matrix, t,-i(W) is the spectrum of y, - " and t._.(w) is the spectrum of tI, - U,. For each of the parametrizations in the previous section, we will derive the asymptotic criterion J!.,(8). We will show the relation with the methods DI, 11 and TS, and we will investigate the bias distribution of the methods. Since most model based controllers do not use a noise model, we assume that no noise model is estimated. In the remaining part of this paper we will omit the constant 1/2r in the expressions for J:"(8), since this does not influence the result of the minimization problem. For readability the w-dependence will be omitted, if it is clear from the context.
u, =
It..! 1
Finally we can write
1 2r tr1" _.. t'_i(w,8) +~t._.(w,8)dw
(18)
+[ I -G(8)
To investigate the characteristics of the proposed closedloop identification procedure for each possible parametrization, we evaluate their asymptotic bias distribution. Making use of Paraeval's relationship as is done by Ljung (1987), the following holds for J1(8) in (12):
=
(17)
that we obtain
t,-i
3. ASYMPTOTIC BIAS DISTRIBUTION
J!.,(8)
lv,
-G(8)
We define
These four possible parametrizations will be further elaborated on in the following section. We will derive frequencydomain expressions for the asymptotic bias distribution, which gives insight in the models that result from the different procedures.
w.p.1
=", -
G( 8)tIt = {Go - G(8)}SoT/' +! I
11, -li,(8)
= tll
J!.,(8)
(16)
Using (6H7) and the parametrization (14) we find that
and li,(8) = G(8), we willahow that we exactly obtain DI. 2. If we chooee a separate parametrization for li and U, 10 u,(P) = S(P)T/, and MP,8) = G(8)S(P)T/" it will be shown that this results in a procedure which is similar to TS, only now the two steps are done simultaneously. 3. If we chooee a joint parametrization for , and U, 10 u,(P) = S(P)T/, and Ji,(P) = GS(P)T/" we can obtain G(9) as the quotient of these models. It will be shown that this procedure is cloeely related to 11. 4. Carrying further the previous step, the signalsli, and U, can be used as instruments in an optimal IV estimation procedure.
lim J1(8)
(15)
Note the similarity with (8b), from which it follows that this choice of parametrization (14) results in the DI procedure. To derive an expression for J:"(8), we have to evaluate t,_i(w) and t._.(w). Clearly
Looking at (6)-(7), and at the identification criterion (12), we see that we can chooee different parametrizations for li, and U" The properties of the model that results from minimizing (12) are determined by this parametrization. The possibilities are listed here, and will be treated more thoroughly in the next section.
N_oo
~ f,(III- ,,(8»'(YI-li,(8» 1=1
The identification criterion (12) has a l~ resemblance with the control criterion (1). Similar criteria have been proposed by Zang et al. (1991), but there it is reshaped into a DI criterion again.
1. If we choose UI
(14b)
,,(8) = G(8)ul
identification criterion Jl (8):
(14a)
1527
~s(P)
(27a)
Go~s(P) + ~G(P)S(8)
(27b)
Two-step variant. In the Two-Step method of Van den Hof and Schrama(1993) the procedure, proposed here, is done in two steps. In this case {3 in (22b) is replaced by f;: 11,(8, f;) = G(8)u,(f;)
(28)
iJ
8 =
1 I::(u,-u,({3»)'(u,-u,({3» N .=1
(29a)
G(8)u,(f;»)'(y, - G(8)u,(f;))
(29b)
In the previous section the parameter estimate f; was estimated by minimizing (34). With this parameter we can calculate the signals 1I.(iJ) and u.(iJ) according to (30), and use them as instruments in an Instrumental Variable (IV) approach. The resulting estimator will be optimal, in the sense that the parameter variance is minimal. For a large number of observations (N .... 00), the function 1(8) in (10) tends to 100(8) with probability one (Ljung, 1987):
argmjn
~ I:: (y, -
(38)
3.4. 11 and u as ln8trumenu
In the first step the signal u,(f;) is calculated. This is equivalent to a reconstruction of the noise-free part U, of U" In the second step this signal u,(f;) is used to identify G(8). These two steps can be written as f; = argmin
with
«:1
100(8)
Note the difference between (29) and the proposed criterion (12). In (29) first f; and then 6 is calculated. Therefore the influence of the error made in the first step on the resulting model is not clear. We are not allowed to write down (26). In the proposed unifying approach f; and 8 are calculated simultaneously, and (26) holds.
= E[(,(iJ)£,(8)] -
i:
~(.(w; iJ, 8) dw
(39)
where (,(iJ) are the instrumental variables {1I,u}, obtained in the previous section, £,(8) ..4(8)y, - B(8)u, is the prediction error, and ~(.(w;/J,8) is their CJ'OSS-spectrum. The dimension of (, is n, x p, where n, is the number of parameters of G(q,8) to be estimated. It can be written as
=
(40)
3.3. Joint Parametrization 0111 and u
with the interpretation that the ith column of (, is given by
The third possibility is to use a joint parametrization of 11 and u: u,({J) =
(41) where K~i)(q, iJ) is a n, x m matrix filter, containing the estimated transfer function matrices GS(f;) and 5(f;). Hence Kc(q,f;) in (40) is a tensor of p n, x m matrix filters. The prediction error can be written as
(30a) (30b)
5({3)",
11,({3) = GS({3)",
The error signals u, - U, and y, - 11, are now given by u, - u,({3)
{So - 5({3)}", + V"., y, - 11.({3)
£,(8)
u, - 5({3)",
= =
y, - GS({3)",
{GoSo - GS(fJ) }'7c + vv.c
100(8) =
~s({3) = So - 5({3) ~Gs({3) = GoSo - GS({3)
(34)
~s({3)
(35a)
~Gs({3)
(35b)
J;"(8) (36)
i:
Q.(8,iJ)~~Q;(8,iJ)dw
(43)
-G(8) ]~~[ I
-G(8) j" dw
(44)
As said before, due to the /I-dependence in the second part of (44), the resulting model will depend on the noise. This is an unwanted property of the DI method: in general we do not want the noise to influence the identification result.
Two-step variant. Instead of calculating G according to (36), it can be done in a (second) identification step. In this step the signals fI,(iJ) and u,(iJ) are used as data signals. The asymptotic model G(8) is then obtained by minimizing Joo (8) with respect to 8:
= tr
[A(8) {Go - G(8)} So]" dw
= [" IGo - G(8)1'ISol'~~ +[ I
Note the similarity with (9d), which shows that parametriation (30) leads to an 11 method.
Joo (8)
J~ Kc(iJ)~~
Choosing parametrization (14) (11 only), we obtain the DI procedure. The asymptotic bias distribution can be seen from J/",:
The model G(f;) is obtained as G(f;) = GS(f;)5- 1 (f;)
(42)
The models that result from the proposed unifying closed-loop identification method depend on the chosen parametrization. The resulting bias distributions are derived in the previous sections, and their interpretation can best be illustrated by assuming a SI80 plant. Note that the conclusions hold for MIMO processes as well.
with Q..({3) = Qw({3) =
Iv,}
3.5. Interpretation
= tr J~ Qw({3)~~Q;({3) + ~~. + ,X~~" dw
= ..4(8).
(33&) (33b)
so that we obtain
+'xQ.. ({3)~~Q:({3)
B(8)UI
Filling in (40) and (42) in (39), we obtain (32)
As before, we define
J;"({3)
= ..4(8)YI -
{{Go -G(8)}So"t+[ I -G(8)
(31)
The separate parametrization of fI and following expression:
1"
u (22) leads to the
"
1 (/I,{3) = 211' 1 tr _" IGoSo - G(8)S(fJ)I'~~ Joo
(37)
+,XISo - 5({3W~~ + ~~. + ,X~~" dw
1528
(45)
The final result a(8) does not depend on the noise. The limit model is asymptotically consistent if the true plant is in the model set. In general this is not the case, but the model error becomes smaller as ISo - S(li) I becomes smaller, independently of the noise.
controlled laboratory setup, have been eV&1uated. Prelimin'U')' experiments with an improved LQ-controller show that the new models are better suited for control design than those obtained with classical closed-loop identification methods.
The same holds for the joint parametrization (30). For SISO systems the criterion J!.o({J) becomes
5. CONCLUSIONS In this paper we have proposed a unifying closed-loop identification method. It includes the classical Direct Identification approach, the Indirect Identification approach and a one-step version of the method proosed by Van den Hof and Scbrama (1993). It is bued on a «eDeralized identification criterion (12). Depending on the parametrization of i and u, different methods are obtained. For all parametrizations the asymptotic bias distribution of the resulting model has been investigated. It appears that if a separate parametrization (22) or a joint parametrization (30) of y and u is chOlleIl, the resulting model is considerably improved, and better suited for control design than when the DI method is US4'!d.
(46)
The errors IGoSo - GS(fJ)1 and ISo - S({J)I might cancel in the finaJ estimate (36), but no theoretical proof is available.
a
In both (45) and (46) it can be seen that prefiltering the data directly influences the bias, whereas in (44) the effect of a prefilter is not clear. The IV-variant, in which the signals y and u are used as instrumental variables, will give asymptotically consistent results as well. However, experience with the IV method has shown that a large number of samples is needed to obtain good results.
A SISO example illustrated the usefulness of the proposed method, and justified the given interpretation. The fact that the propolled closed-loop identification method unifies several already known identification methods, makes it possible to further compare these methods, and investigate their behaviour more thoroughly.
4. SIMULATION RESULTS To illustrate the ueefulness of the proposed closed-loop identification methods, they are applied to a simple SISO PIDcontrolled process. The transfer function Go of the process is
3.lq3 + 3.6325q2 - l.9090q + 0.3796 qS _ 3.27q4 + 4.098q3 - 2.3317 q2 + 0.S081q q4 _
6. REFERENCES
(47)
Bitmead, R.R., M. GeVeI'll and V. Wertz (1990). Adaptive Optimal Control - the thinking man'" GPC, Prentice Hall, Australia. Gevers, M. (1993). Towards a joint design of identification and control? In: E88ays on control: perspectives in the theory and its applications (H.L. Trentelman and J.C. Willems, Eds.), pp. 111-152. Birkhauser, Boston, MA. Gustavsson, I., L. Ljung and T. SOderstrom (1977). Identification of processes il1 closed loop - Identifiability and accuracy aspects. Automatica, 13, 59-75. Ljung, L. (1987). System Identification: theory for the U8er. Prentice-Hall, Englewood Cliffs, NJ. SOderstrom, T., I. Gustavsson and L. Ljung (1975). Identifiability conditions for linear systems operating in closed loop. Int. J. Control. 21,243-255. SOderstrom, T., P. Stoica and E. Trulsson (1987). Instrumental variable methods for closed loop systems. Prepr. of the 10th IFAC World Congrus on Automatic Control, Munich, FRD, pp. 364-369. Van den Hof, P.M.J. and R.J.P. Schrama (1993). An indirect method for transfer function estimation from closed loop data. Automatica, 29, 1523-1527. Van Osch, J.E.F. (1992): Identification of multitlariable proceue.5 operating under feedbac/c, MSc. thesis report, no. A92.053(615), Delft University of Technology, Control Laboratory, The Netherlands. Zang, Z., R.R. Bitmead and M. Gevers (1991). Iterative model refinement and control robU8tne.5S enhancement. Report no. 91.137, Center for Systems Eng. and AppJ. Mech., Univ. Cath. de Louvain-La-Neuve, Belgium.
The noise signals W." and W ..,' (see Fig. 1) are white noise, coloured by different filters. The loop is excited via the external input signal e,. The fifth-order plant is approximated by fourth-order models. The resulting Nyquist diagrams are shown in Fig. 2.
Ur--~--------~-----, ~
. .
.....
~
.
.
; .
~
,!,
.
; .
~
.
i y..... :
·1 -..
.
~
..
~
L...... . .
..
i
.~.~ .. ;,:~~.; ..•......... :.....
·····L
·~I!---...,~--O~---1'.cu:---~--..,.U:---~
ig.
2. Nyquiat dia«rama of the models: original (eolid), DI (14) (dotted), 8epara1e parametriaatioD (22) (dub-dotted), joint parametrisatioD (30) (daabed), IV (39) (bold-dotted)
'he interpretation in eection 3.5 is confirmed by Fig. 2. The 1I method does not give a sood estimate of the true plant. 'he other two parametrizations, (22) and (30), give good !Suits, as well as IV. his also holds for MIMO processes, as shown by Van Osch .992), in which several MIMO experiments, including a
1529
CLOSED-LOOP IDENTIFICATION METHODS FOR LQ CONTROL DESIGN A.C. VAN DER KLAUW·,t, J.E.F. VAN OSCH· and P.P.J. VAN DEN BOSCH·· °Delft Uaiwm', 0/ Tecltaolon. De,.. 0/ Eleetric.l Ea,., Coa'rol Ldo,..'orr. P.D. Bft SO'l. liDO CA DelfI. Tle NetAerl..4.
.oEiaaOtlefl Uaitlerftt, 0/ Tecltao/on. De,.. 0/ Eleetric.l Ea,.• M_remea' MB Eiaao"ea. Tle NetAerl.ah
.a~
Coa'rol Sedioa. P.D. Bn Sl'. S600
Abatrae&. ID CODDedion with the current intereat in the interae\ion between identification and control deaip, a clc.ed-loop identification method is propoaed that take- into account a aubeequent LQ control deaip mp. It appears that tbia method unifiea emtin& clc.ed-loop identification acheme-. Hence it facililatea comparison of tbae acheme-. The propert.iea of the rsultinl model depend OD the chc.en parametriaation. It is ahown that the npt choice of parametrisation reau1ta in a model, of which the biu distribution is independent of the noi8e, which is an advant&«e with reapect to aome d_ca1 approachea. The method is illuatrated with a aimulation eumple.
Key Worda. Clc.ecl-loop ayItema, multivariable ayItema, identification for control, eaiimation theory, parameter MtimatioD. ~ identificatioD. .
loop excitation signals r, (reference signal) or e, (extemal input). The process output is 11" and the process input is u,. The dimensions of the signals are: r"11" W",: P x 1; e" u" W ..,': m x 1. The transfer function matrices have appropriate dimensions.
1. INTRODUCTION
The problem of identifying multivariable processes operating under feedback has attracted much attention off late (Bitmead et al., 1990; Geven, 1993; Van den Hof and Schrama, 1993; Zang et al., 1991). This has been inspired by the wish to make better use of the interaction between identification and control design. The key idea is that in the identification step the control step should already be taken into account. To improve a controller, it is argued by e.g. Gevers (1993) that the model should be obtained from closed-loop data. Another reason for doing identification under feedback is th!,-t unstable systems can be stabilized. For the process industry an additional reason is that output quality can be maintained during the experiment, thus saving costs.
We assume that the controller {C" C,} needs improvement, and that it is obtained by minimizing the LQ criterion JC(C"C,):
JC(C" C.) =
~ t {(r, -;,(Cb C.; 8»'(r, -;,(C" C.i 8» &=1
+AU,(Cb C.J)'u,(C" C.; 8)}
where' denotes the transpose of a matrix, and where A is a positive constant. Note that the controller {Cb C.} is calculated on the basis of a model G(8) of the real plant Go.
This paper is concerned with identifying a model of a plant, that is operating in closed loop. In Fig. 1 a typical closedloop situation is depicted. The plant is denoted by Go(q),
The problem we address in this paper is to obtain a model G(q,8) of the plant Go(q), from N measurements of 11" u" re and e" that is suitable for control design according to (1). The model is parametrized with parameter vector 8; and 8 is the estimated parameter vector. It is explicitly assumed that the process Go is not in the model set, which implies that we do not only want the identification procedure to give consistent estimates asymptotically, but we also need tunable expressions for the asymptotic bias distribution. Problems are caused by the correlation between the process input and the noise. In oven-loop identification this correlation is not present. However, from Fig. 1 it follows that this is not true in closed·loop situations.
Fil. 1. Typical cIoIed-loop COllfiIuration
where q-I is the well-known delay operator. For readability q is often omitted in the equations in this paper. The controller is a two-desree of freedom controller, denoted by CJ(q) and C.(q). All disturbances are lumped into the noise signals w", and w..,. It is supposed that the loop is excited by at least one of the • ne reoearda of A.C.
ftII
(1)
In the following section we will briefly trea.t some classical closed-loop identification procedures, that are not tailored for control design. We then come to the formulation of an identification criterion, suitable for LQ control design.
cIer IQuw ia made poMiblo by ~e DlIkh Tedl-
aaiocY FoaadUioa ucler Grut STW DEL92.1nll
1525
is obtained with a Prediction Error Method (PEM) as:
Subsequently, in section 3 we will show that the proposed method also includes several classical closed-loop identification schemes. The properties of the asymptotic models depend on the particular parametrization, which is illustrated by deriving the asymptotic bias distribution for different parametrizations. A simulation example illustrates the method, and we draw some conclusions in section 5.
[I Go] Wl
,
ti = arg min In(I) In(8) =
(2)
/(8)
=
C/~,(w)Ci
+C/~"(w)
(3)
Ul
(5)
Ut 1/t
where· denotes the conjugated transpOlle of a complex matrix, and with ~, the spectrum of rh ~. the spectrum of eh and ~ .. and ~ .. their cross-spectra. We now define the dOlled.loop traM/er junction go as:
L
' '} N1 N (I {A(I)Yl - B(8)ul
(lOa) (lOb)
=
Ul+V...t SO"'t GOUl +Vr,l
(Ba) (Bb) (Bc)
In the first step an estimate S of the sensitivity function So is calculated. With this estimate the noise-free signal Ul is calculated. Observing that in (Bc) the signal Ut and the noise Vr,l are uncorreiated, an estimate G can be obtained from open-loop calculations. Although this method asymptotically yields a consistent estimate, it is not clear how the first step influences the second step. A thorough analysis is not yet available. More about this method will be said in section 3.2.
(6)
so that finally we have Zl = .il + VI = YO"'1 + VI
=
= 0)
Recently a new method was proposed by Van den Hof and Schrama (1993), called the TIDO-Step method (TS). For SISO systems, an expreaion for the asymptotic bias distribution was derived, in the case where the system is not in the model set. The method is based on the following relations:
will be used later, and is given
+ ~.(w) + ~ .. (w)Ci
(9c) (9d)
=
(4)
~,,(w)
(9b)
tsl
with G(8) ..4-1 (I)B(I). From (8)-(10) it is clear that the classical methods are variatfons of the well-known open-loop methods. For closed-loop systems they have two disadvantages. First of all, if the process is not in the model set, the resulting model is biased, and the bias depends on the noise. Secondly, they do not take into account the purpose of the model: control design.
"'t
~,,(w),
(rl - YI(I/»'(rl - Yl(l/»
(9a)
1=1
In general the controller {C" C.} is known, and Cl is stable. Hence we can define a loop excitation signal as
The spectrum of"'h by
t
~1(J(8)
=
= [ ~o ] SO{Clrl +~} + VI
(Se)
The plant model G(ti) is obtained by solving (9d). Using IMtrumental Varia61e (IV) techniques, a model of the plant can be obtained as well (SOderatrom, 1987). However, the construction of the instruments is more involved than in the open-loop case, since the input Ul is correlated with the noise. Provided appropriAte instruments (I can be found, the parameter estimate ti is obtained as
=
Zl
~
il(8) = GS(8)"'1 GS(ti) = G(ti)[I + C.G(ti»)
Let us define z(t) (r: u:)'. The first part of (2) can be seen as the noise-free part of z(t), and will be denoted by .il . The second part of (2) is the noise contribution, denoted by VI (V~.l V~.l)'. Furthermore the loop aeMitit7ity junction So is defined as So [I + C.GO)-I. We then obtain the following:
=
(8b)
The disadvantage of this method is the dependence of ti on the noise. This will be shown in eection 3. With Indirect Identification (11) first a model of the cloeed loop is estimated, between the loop excitation signal "" and the output rl' Accurate knowledge of the controller then yields a model of the plant:
[Cl I] ( :: ) l
(YI - Yl(8»'(rl - Yl(I/»
iM) = G(8)uI
Let us first derive the equations that describe the loop behaviour in Fig. 1. Defining the noise vector Wl as Wt = (W~.l w~,t)', the equations can be written as:
[_~.] [I + GoC.r
~t
(Sa)
1=1
In this eection we will briefly treat some classical cloeedloop identification methods, that methods do not take into account that the model will be used for control desipl. Therefore an extension is proposed, that is clOllely related to LQ-type controllers. Within this method we can vary the parametrlzation, which influences the properties of the resulting model. This will become clear in the sequel.
+
= argminJOI(8)
JOI(8)
2. CLOSED-LOOP IDENTIFICATION
( ~:) = [~o] [I + C.Gor l
,
ti
(7)
Without taking into account control design, the closed-loop identification problem is the problem of obtaining a model G(ti) of Go from cloeed-loop d&ia. In recent years, eeveral procedures for cloeed-loop identification have been proposed (e.g. Gustavsson et al., 1977; SOderatrom et al. 1975, 1987). The mostly used method is probably Direct Identification (DJ). Measurements of both rl and Ul are taken, and procesaed without taking into ac: count the presence of feedback. The parameter estimate 8
To improve the identification results, in this paper we proa cloeed-loop identification method that takes into account both LQ control desipl and undermodelling. It appears to unify DJ, 11 and TS, where the two steps in the TS method are taken together, so that analysis is now p0ssible. The main ingredie!lt i~ the d~tion of the global pOlIe
1526
~ f, {(YI -li,)'(1I1 -li,)
J1(8) =
The criterion (12) then becomes
1:1
+~( tI, -
UI)'( tI,
-
UI)}
J1(8) =
(12)
t._.(w) = 0
~G(8) = Go - G(8) 10
=
~(8)Sot~{~G(8)So}O
J!.,(8)
1 -G(8)
It.. ! I
1°
(19)
-G(8)
1° dw
(20)
with (21) Minimization of J:"(8) will result in a small model error ~G(8), but the model quality is dependent on the noise, due to the 8-dependence of the noise contribution in (20). This effect is of course unwanted.
3.2. Separate Parametrization of' and
u
The second possible parametrization is
(13&)
u,(P) =
,,(8, P) =
S(P)T/,
G(8)ul(P)
(22&) (22b)
To derive J:"(8, P), we will calculate tI, - U, and ", - ',. We see from (6)-(7) and the parametrization (22) that
(13b)
tll - iI,(P)
tI, -
S(P)T/,
= {So - S(P)}"d v." Y, - ,,(8,P)
=
(23)
Y, - G(8)S(P)T/1 {GoSo - G(8)S(P)}T/' + v",
(24)
We define ~s(P) = So - S(P) ~(8) 10
=
(25b)
that we obtain J!.,(8,P)
3.1. Parametrization of' Only
(25&)
Go - G(8)
= tr [ Q,(8,P)t~Q;(8,P)
+.\Q.(P)t~Q:(P)
+ t .. + .\t.. dw
(26)
with
The first parametrization is tI,
-G(8)
= tr J~ Q,(8)t~Q;(8)
+!
where w.p. 1 means tDith probability one, tr is the trace of a matrix, t,-i(W) is the spectrum of y, - " and t._.(w) is the spectrum of tI, - U,. For each of the parametrizations in the previous section, we will derive the asymptotic criterion J!.,(8). We will show the relation with the methods DI, 11 and TS, and we will investigate the bias distribution of the methods. Since most model based controllers do not use a noise model, we assume that no noise model is estimated. In the remaining part of this paper we will omit the constant 1/2r in the expressions for J:"(8), since this does not influence the result of the minimization problem. For readability the w-dependence will be omitted, if it is clear from the context.
u, =
It..! 1
Finally we can write
1 2r tr1" _.. t'_i(w,8) +~t._.(w,8)dw
(18)
+[ I -G(8)
To investigate the characteristics of the proposed closedloop identification procedure for each possible parametrization, we evaluate their asymptotic bias distribution. Making use of Paraeval's relationship as is done by Ljung (1987), the following holds for J1(8) in (12):
=
(17)
that we obtain
t,-i
3. ASYMPTOTIC BIAS DISTRIBUTION
J!.,(8)
lv,
-G(8)
We define
These four possible parametrizations will be further elaborated on in the following section. We will derive frequencydomain expressions for the asymptotic bias distribution, which gives insight in the models that result from the different procedures.
w.p.1
=", -
G( 8)tIt = {Go - G(8)}SoT/' +! I
11, -li,(8)
= tll
J!.,(8)
(16)
Using (6H7) and the parametrization (14) we find that
and li,(8) = G(8), we willahow that we exactly obtain DI. 2. If we chooee a separate parametrization for li and U, 10 u,(P) = S(P)T/, and MP,8) = G(8)S(P)T/" it will be shown that this results in a procedure which is similar to TS, only now the two steps are done simultaneously. 3. If we chooee a joint parametrization for , and U, 10 u,(P) = S(P)T/, and Ji,(P) = GS(P)T/" we can obtain G(9) as the quotient of these models. It will be shown that this procedure is cloeely related to 11. 4. Carrying further the previous step, the signalsli, and U, can be used as instruments in an optimal IV estimation procedure.
lim J1(8)
(15)
Note the similarity with (8b), from which it follows that this choice of parametrization (14) results in the DI procedure. To derive an expression for J:"(8), we have to evaluate t,_i(w) and t._.(w). Clearly
Looking at (6)-(7), and at the identification criterion (12), we see that we can chooee different parametrizations for li, and U" The properties of the model that results from minimizing (12) are determined by this parametrization. The possibilities are listed here, and will be treated more thoroughly in the next section.
N_oo
~ f,(III- ,,(8»'(YI-li,(8» 1=1
The identification criterion (12) has a l~ resemblance with the control criterion (1). Similar criteria have been proposed by Zang et al. (1991), but there it is reshaped into a DI criterion again.
1. If we choose UI
(14b)
,,(8) = G(8)ul
identification criterion Jl (8):
(14a)
1527
~s(P)
(27a)
Go~s(P) + ~G(P)S(8)
(27b)
Two-step variant. In the Two-Step method of Van den Hof and Schrama(1993) the procedure, proposed here, is done in two steps. In this case {3 in (22b) is replaced by f;: 11,(8, f;) = G(8)u,(f;)
(28)
iJ
8 =
1 I::(u,-u,({3»)'(u,-u,({3» N .=1
(29a)
G(8)u,(f;»)'(y, - G(8)u,(f;))
(29b)
In the previous section the parameter estimate f; was estimated by minimizing (34). With this parameter we can calculate the signals 1I.(iJ) and u.(iJ) according to (30), and use them as instruments in an Instrumental Variable (IV) approach. The resulting estimator will be optimal, in the sense that the parameter variance is minimal. For a large number of observations (N .... 00), the function 1(8) in (10) tends to 100(8) with probability one (Ljung, 1987):
argmjn
~ I:: (y, -
(38)
3.4. 11 and u as ln8trumenu
In the first step the signal u,(f;) is calculated. This is equivalent to a reconstruction of the noise-free part U, of U" In the second step this signal u,(f;) is used to identify G(8). These two steps can be written as f; = argmin
with
«:1
100(8)
Note the difference between (29) and the proposed criterion (12). In (29) first f; and then 6 is calculated. Therefore the influence of the error made in the first step on the resulting model is not clear. We are not allowed to write down (26). In the proposed unifying approach f; and 8 are calculated simultaneously, and (26) holds.
= E[(,(iJ)£,(8)] -
i:
~(.(w; iJ, 8) dw
(39)
where (,(iJ) are the instrumental variables {1I,u}, obtained in the previous section, £,(8) ..4(8)y, - B(8)u, is the prediction error, and ~(.(w;/J,8) is their CJ'OSS-spectrum. The dimension of (, is n, x p, where n, is the number of parameters of G(q,8) to be estimated. It can be written as
=
(40)
3.3. Joint Parametrization 0111 and u
with the interpretation that the ith column of (, is given by
The third possibility is to use a joint parametrization of 11 and u: u,({J) =
(41) where K~i)(q, iJ) is a n, x m matrix filter, containing the estimated transfer function matrices GS(f;) and 5(f;). Hence Kc(q,f;) in (40) is a tensor of p n, x m matrix filters. The prediction error can be written as
(30a) (30b)
5({3)",
11,({3) = GS({3)",
The error signals u, - U, and y, - 11, are now given by u, - u,({3)
{So - 5({3)}", + V"., y, - 11.({3)
£,(8)
u, - 5({3)",
= =
y, - GS({3)",
{GoSo - GS(fJ) }'7c + vv.c
100(8) =
~s({3) = So - 5({3) ~Gs({3) = GoSo - GS({3)
(34)
~s({3)
(35a)
~Gs({3)
(35b)
J;"(8) (36)
i:
Q.(8,iJ)~~Q;(8,iJ)dw
(43)
-G(8) ]~~[ I
-G(8) j" dw
(44)
As said before, due to the /I-dependence in the second part of (44), the resulting model will depend on the noise. This is an unwanted property of the DI method: in general we do not want the noise to influence the identification result.
Two-step variant. Instead of calculating G according to (36), it can be done in a (second) identification step. In this step the signals fI,(iJ) and u,(iJ) are used as data signals. The asymptotic model G(8) is then obtained by minimizing Joo (8) with respect to 8:
= tr
[A(8) {Go - G(8)} So]" dw
= [" IGo - G(8)1'ISol'~~ +[ I
Note the similarity with (9d), which shows that parametriation (30) leads to an 11 method.
Joo (8)
J~ Kc(iJ)~~
Choosing parametrization (14) (11 only), we obtain the DI procedure. The asymptotic bias distribution can be seen from J/",:
The model G(f;) is obtained as G(f;) = GS(f;)5- 1 (f;)
(42)
The models that result from the proposed unifying closed-loop identification method depend on the chosen parametrization. The resulting bias distributions are derived in the previous sections, and their interpretation can best be illustrated by assuming a SI80 plant. Note that the conclusions hold for MIMO processes as well.
with Q..({3) = Qw({3) =
Iv,}
3.5. Interpretation
= tr J~ Qw({3)~~Q;({3) + ~~. + ,X~~" dw
= ..4(8).
(33&) (33b)
so that we obtain
+'xQ.. ({3)~~Q:({3)
B(8)UI
Filling in (40) and (42) in (39), we obtain (32)
As before, we define
J;"({3)
= ..4(8)YI -
{{Go -G(8)}So"t+[ I -G(8)
(31)
The separate parametrization of fI and following expression:
1"
u (22) leads to the
"
1 (/I,{3) = 211' 1 tr _" IGoSo - G(8)S(fJ)I'~~ Joo
(37)
+,XISo - 5({3W~~ + ~~. + ,X~~" dw
1528
(45)
The final result a(8) does not depend on the noise. The limit model is asymptotically consistent if the true plant is in the model set. In general this is not the case, but the model error becomes smaller as ISo - S(li) I becomes smaller, independently of the noise.
controlled laboratory setup, have been eV&1uated. Prelimin'U')' experiments with an improved LQ-controller show that the new models are better suited for control design than those obtained with classical closed-loop identification methods.
The same holds for the joint parametrization (30). For SISO systems the criterion J!.o({J) becomes
5. CONCLUSIONS In this paper we have proposed a unifying closed-loop identification method. It includes the classical Direct Identification approach, the Indirect Identification approach and a one-step version of the method proosed by Van den Hof and Scbrama (1993). It is bued on a «eDeralized identification criterion (12). Depending on the parametrization of i and u, different methods are obtained. For all parametrizations the asymptotic bias distribution of the resulting model has been investigated. It appears that if a separate parametrization (22) or a joint parametrization (30) of y and u is chOlleIl, the resulting model is considerably improved, and better suited for control design than when the DI method is US4'!d.
(46)
The errors IGoSo - GS(fJ)1 and ISo - S({J)I might cancel in the finaJ estimate (36), but no theoretical proof is available.
a
In both (45) and (46) it can be seen that prefiltering the data directly influences the bias, whereas in (44) the effect of a prefilter is not clear. The IV-variant, in which the signals y and u are used as instrumental variables, will give asymptotically consistent results as well. However, experience with the IV method has shown that a large number of samples is needed to obtain good results.
A SISO example illustrated the usefulness of the proposed method, and justified the given interpretation. The fact that the propolled closed-loop identification method unifies several already known identification methods, makes it possible to further compare these methods, and investigate their behaviour more thoroughly.
4. SIMULATION RESULTS To illustrate the ueefulness of the proposed closed-loop identification methods, they are applied to a simple SISO PIDcontrolled process. The transfer function Go of the process is
3.lq3 + 3.6325q2 - l.9090q + 0.3796 qS _ 3.27q4 + 4.098q3 - 2.3317 q2 + 0.S081q q4 _
6. REFERENCES
(47)
Bitmead, R.R., M. GeVeI'll and V. Wertz (1990). Adaptive Optimal Control - the thinking man'" GPC, Prentice Hall, Australia. Gevers, M. (1993). Towards a joint design of identification and control? In: E88ays on control: perspectives in the theory and its applications (H.L. Trentelman and J.C. Willems, Eds.), pp. 111-152. Birkhauser, Boston, MA. Gustavsson, I., L. Ljung and T. SOderstrom (1977). Identification of processes il1 closed loop - Identifiability and accuracy aspects. Automatica, 13, 59-75. Ljung, L. (1987). System Identification: theory for the U8er. Prentice-Hall, Englewood Cliffs, NJ. SOderstrom, T., I. Gustavsson and L. Ljung (1975). Identifiability conditions for linear systems operating in closed loop. Int. J. Control. 21,243-255. SOderstrom, T., P. Stoica and E. Trulsson (1987). Instrumental variable methods for closed loop systems. Prepr. of the 10th IFAC World Congrus on Automatic Control, Munich, FRD, pp. 364-369. Van den Hof, P.M.J. and R.J.P. Schrama (1993). An indirect method for transfer function estimation from closed loop data. Automatica, 29, 1523-1527. Van Osch, J.E.F. (1992): Identification of multitlariable proceue.5 operating under feedbac/c, MSc. thesis report, no. A92.053(615), Delft University of Technology, Control Laboratory, The Netherlands. Zang, Z., R.R. Bitmead and M. Gevers (1991). Iterative model refinement and control robU8tne.5S enhancement. Report no. 91.137, Center for Systems Eng. and AppJ. Mech., Univ. Cath. de Louvain-La-Neuve, Belgium.
The noise signals W." and W ..,' (see Fig. 1) are white noise, coloured by different filters. The loop is excited via the external input signal e,. The fifth-order plant is approximated by fourth-order models. The resulting Nyquist diagrams are shown in Fig. 2.
Ur--~--------~-----, ~
. .
.....
~
.
.
; .
~
,!,
.
; .
~
.
i y..... :
·1 -..
.
~
..
~
L...... . .
..
i
.~.~ .. ;,:~~.; ..•......... :.....
·····L
·~I!---...,~--O~---1'.cu:---~--..,.U:---~
ig.
2. Nyquiat dia«rama of the models: original (eolid), DI (14) (dotted), 8epara1e parametriaatioD (22) (dub-dotted), joint parametrisatioD (30) (daabed), IV (39) (bold-dotted)
'he interpretation in eection 3.5 is confirmed by Fig. 2. The 1I method does not give a sood estimate of the true plant. 'he other two parametrizations, (22) and (30), give good !Suits, as well as IV. his also holds for MIMO processes, as shown by Van Osch .992), in which several MIMO experiments, including a
1529