O_hp. W_e. d_ohp PWR. U_cex. Q_ex. d_ucex. Primary and. N_cd. Secondary Circuit. K_tb. W_ref. K_cd. turbine control valve. secondary circuit TURBINE

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1 A Comparison Between Model Reduction and Contrler Reduction: Application to a PWR Nuear Plant y Beno^t Codrons,Pascale Bendotti, Clement-Marc Falinower and Michel Gevers CESAME, Universite Cathique de Louvain, - avenue G. Lema^tre, B-8 Louvain-la-Neuve, Belgium Codrons@csam.u.ac.be, Gevers@csam.u.ac.be Electricite defrance, Direction des Etudes et Recherches, Quai Watier, F-8 Chatou, France Pascale.Bendotti@edf.fr, Clement-Marc.Falinower@edf.fr Abstract The most assical way of obtaining a low-order modelbased contrler for a high-order system is to apply reduction techniques to an accurate high-order model or contrler of the plant. An alternative isto use identication for contr with a low-order model set. In [], we compared model reduction, both in open loop and in osed loop, with osed-loop identication of a low-order model. Both methods were applied to the design of a contrler for the secondary circuit of a nuear Pressurized Water Reactor, leading to the conusions that system identication is a viable alternative to model reduction, and that the key feature for a successful contr design is not so much the choice between order reduction or identication, but between open-loop and osed-loop techniques. In the present paper, we go further in this study and we compare model reduction with contrler reduction for the same PWR system. We show that osed-loop techniques are more powerful than open-loop ones, and that contrler reduction gives better results than model reduction when appropriate frequency weightings are chosen. Introduction There are several ways of obtaining low-order contrlers for high-order systems. One line of thinking is to rst obtain a very accurate high-order model and then apply reduction techniques to this model or to a high-order contrler computed from that model. There is an extensive literature on this subject. One of the important theoretical messages of this literature is that, if the ultimate objective isthelow-order contrler (rather than the low-order model), then it is y The authors acknowledge the Belgian Programme on Interuniversity Pes of Attraction, initiated by the Belgian State, Prime Minister's Oce for Science, Technogy and Culture. The scientic responsibility rests with its authors. essential that the osed-loop performance objective be incorporated in the reduction technique. This is typically achieved by specic frequency weightings that translate these osed-loop objectives in the model or contrler reduction criterion. The recent research on identication for contr has promoted the idea that one can, alternatively, obtain a low-order model directly by osed-loop identication, where the identication criterion takes account of the contr performance objective. The approach based on model reduction (both in open loop and in osed loop) was compared to that based on osed-loop identication in []. For that purpose, both methods were applied to the design of a low-order contrler for a realistic model of order of a Pressurized Water Reactor (PWR) nuear power plant. The main contribution of that paper was to add insight to the ongoing debate about identication for contr. Two important ndings were produced. Firstly, osed-loop identication proved to be a viable alternative to model reduction: we produced a -th order contrler obtained through osed-loop model reduction techniques that achieved a required level of performance and we showed that the same performance was achieved by a -th order contrler obtained from a low-order model of the PWR directly identied via a contr-oriented identication criterion. In both cases the same LQG criterion was used for the design. Secondly we showed that, whether the route to a low-order contrler is via model reduction techniques or via identication of a low-order model, the key to the success of the operation is to inject weightings that reect the osed-loop performance objectives. In the present paper, we focus on the approach that was not considered in [], namely computation of a fullorder contrler flowed by contrler reduction. We Taking account of LQG weighting lters that are actually part of the designed contrlers, their order was in fact. p.

2 compare this with the model reduction approach considered in []. Therefore, the same -nd order model of the PWR is used. It is rst reduced to a model of order, for reasons explained in Section 5. For this -rd order model, a full LQG contrler of order is computed and then reduced to order. We show that its performance is similar to that of the -th order contrler obtained via reduction of the model to order flowed by contr design with the same LQG criterion. But in addition, we show that if we push down the reduction to the lowest possible order, the results obtained by osed-loop contrler reduction can be signicantly better than those obtained by osedloop model reduction or open-loop contrler reduction, provided appropriate weightings are used. The outline of the paper is as flows. In Section we sketch the modelling of the PWR plant, while Section gives a description of the contr problem and the contr design procedure. Section reviews the coprime factor model order reduction procedure already used in [], rst in open loop, then when osed-loop considerations are taken into account. It shows how the contr objective can be used to select an adequate frequency weighting for the reduction. Section 5 describes the procedure used to reduce the contrler in an open-loop as well as in a osed-loop sense. The performance on the actual system of the low-order contrlers obtained in Sections and 5 are compared in Section. Finally, some conusions are drawn in Section. W_e REACTOR Q_ex N_cd contr rod system STEAM GENERATOR to the pressurizer primary circuit reactor coant pump secondary circuit feedwater contr valve turbine contr valve TURBINE CONDENSER feedwater pump Figure : PWR plant description PWR Primary and Secondary Circuit P O_hp U_cex d_ohp d_ucex Modelling of the PWR A realistic nonlinear simulator, based on a rst principles' model describing both primary and secondary circuits of the PWR (See Figure ), has been developed at Electricite de France (EDF). It inudes all local contrlers invved in both primary and secondary circuits. In this paper, we focus on the behavior of the plant around a xed operating point corresponding to 95% of maximum operating power. This results in a high (-nd) order model P, which inudes the dynamics of the primary and secondary circuits and of all local contrlers, except some specic contrlers of the secondary circuit that we want to redesign these are denoted tb and cd in Figure. They contr the electrical power and the condenser water level, respectively, and their structures are very simple: tb is a PI contrler acting on the dierence between its two inputs, while cd is a second order two-input-one-output contrler which inudes an integrator. For the sake of simplicity, tb and cd will both be called \PID" contrlers in the sequel. In Figure, is the electrical power produced by W_ref _tb _cd Figure : Interconnection of P with the PID contrlers. the plant, contrled to flow the demand W ref of the network, and directly related to the steam ow inthe turbine, which depends on the high pressure turbine contr valve aperture - see Figure is the extraction water ow, and the water level in the condenser (both are related to the locally contrled speed of the feedwater pump and to the extraction valve aperture ). d Ohp and d Ucex represent additive terms on the contr inputs that can be either disturbances, or excitations for identication purposes. Obviously, there is a strong coupling between and on the one hand, and between and on the other hand, which would explain the structure of the present PIDcontrlers. However, the contr performance might beenhancedby taking the cross-couplings into account. p.

3 Contr design strategy Our goal is to redesign contrlers for the electrical power contr in the secondary circuit, i.e. to replace the present PID contrlers tb and cd by a single multivariable contrler in order to achieve a better performance. The chosen contr design is a Linear Quadratic Gaussian (LQG) contrler computed from a reduced order model ^Pr of the plant P (or from that plant itself). The contr objective is to use the feedwater tank, rather than the contr rods in the primary circuit, to absorb the fast and medium range variations in the power demands by acting on the valve apertures. The contrler will have to ensure that the electrical power supply flows accurately the reference signal W ref, and to regulate the condenser water level around its nominal value: see Figure. Also, it will have to reject possible disturbances acting on the system at the inputs d Ohp and d Ucex. The validation will be done with step signals on W ref, d Ohp and d Ucex. In order to remain consistent in the comparativestudy, the same LQG criterion is used with each (reduced order) model: J LQG (u) = Z ; y T filt Qy filt + u T Ru dt () where u = T is the contr vector and y filt = T F (s)( ; W ref ) F (s) is the contrled output vector ltered through a lter F (s) = +=s to ensure a zero static error. This lter is then connected to the corresponding inputs of the designed contrler. Since the main goal is to contr (the regulation of being only a secondary requirement), more weight is put on the electrical power tracking error than on the condenser water level in J LQG. On the other hand, since the nominal value of is. while it is for, times more weight isputon to ensure a correct scaling. The chosen weighting matrices are Q = ; R = : () These weightings have proved very satisfactory. For the design of the alman lter, the external signals d Ohp, d Ucex and W ref, which are in the low The weighting matrices and lters used here are dierent from those used in []. However, we have veried that all conusions and observations made in [] would remain valid with the weightings used here. frequency ranges, are modeled as independent Gaussian white noises ltered through a low-pass lter N (s) = = (s + :). Since the external signals have a typical amplitude of for d Ohp and W ref, and of. for d Ucex, their covariance matrix is chosen as Q n = diag ( : ). In order to ensure a good rl-o at high frequency, the measurement noise is parametrized as a Gaussian white noise with large covariance R n = diag ( ) (remember that is measured and used for state estimation, although it is not regulated, which iswhy R n is ). The presence of the lters F (s) (twice) and N (s) ( times) in the model used for the design will yield a contrler with order equal to that of this design model, i.e. the order of ^Pr plus 5. Furthermore, F (s) must be explicitely added to the designed contrler before it can be used with ^Pr or P, and the total order of the contrler will therefore nally be that of ^Pr plus. This will determine the order of ^Pr if a contrler of some xed order is desired. First approach to a low-order contrler: model reduction flowed by contr design Our rst approach consists in reducing the order of the -nd order plant modelp to a model of order (for the sake of comparison with the model identied in Section of []), and to use this resulting low-order model to design a contrler. Since the system P inudes unstable modes, a straight balanced truncation is not achievable. Therefore we use a factorization method in which the system transfer function is factored into stable coprime factors, as proposed by Meyer []. This section reviews material already presented in [] however, Subsection. has been re-written such that the new results of Subsection 5. can be derived from the same expressions, making them much easier to understand.. Open-loop coprime factor reduction Since the unstable system under consideration is detectable, we can construct a stable left coprime factorization (LCF) ~ N ~ M such that P = ~M ; ~ N. We refer the reader to [] for more details on how such a factorization can be obtained. The coprime factors can be normalized, meaning that ~N ~ N? + ~ M ~ M? = I where X? (s) =X T (;s). Since N ~ M ~ is stable, it can be reduced using standard balanced truncation. One advantage of using normalized coprime factors is that the error between the full- and reducedp.

4 order models can then be interpreted in the graph metric or gap metric [,, ]: the error is an upper bound on the distance between the graphs of the full and reduced order models. Let Nr ~ Mr ~ denote the reduced LCF realization. To make the reduction process useful, we must ensure that the reduced factors Nr ~ and ~M r have the same denominator, and that their realizations have the same state, so that the order of the reduced model ^Pr = M ~ ; r ~N r is that of Nr ~ and Mr ~ rather than the sum of these. We apply the LCF reduction method to P. We denote by ^P r the reduced order model obtained by truncating all but the rst r singular values. Recall that for such model an upper bound on the committed approximation error in the H norm is given by twice the sum of the truncated Hankel singular values (HS): kp ; ^P r k P i=r+ i. We have observed in [] that the model ^P produces a destabilizing LQG ^P contrler of order,, when applied to the true system, and that it is necessary to limit the reduction at the order ( ^P ) to obtain a stabilizing contrler. ^P The latter is denoted 9 and is of order 9.. Closed-loop coprime factor reduction Let us rewrite the two PID contrlers tb and cd of Figure as a single contrler PID. The idea of osed-loop model reduction is to compute the reduced-order model ^Pr that ensures the best possible matching between the osed-loop transfer functions T (P PID )andt ^Pr PID. The osed-loop system can be redrawn as in Figure, where y = T, r = Wref T, u = ;Ohp ; T and d = T d Ohp d Ucex. Here, denotes a ctitious nil signal. Note that a ctitious output has been added to P to make the dimensions of all vectors compatible. u y = = PID (I + P I PID ) ; P I d r U ; N ~ M ~ d r () where = ~ N U + ~ M can be made equal to I for some choice of U and (to simplify the notations, we shall assume that in the sequel). Minimizing the approximation error between the two osed-loop transfer functions, in which PID remains unchanged, is then equivalent with minimizing a frequency weighted difference between the LCF's of P and ^Pr : min ^P r T (P PID ) ; T ^Pr PID m () ; min Wout ~N M ~ ; ~Nr Mr ~ Win : f ~ N r ~ M rg The weightings W in and W out are chosen in order to take account ofthe entries of y, r and d that are important for the contr design, namely,, d Ohp, d Ucex and W ref. Considering () and the denition of y, r and d yields and W out = W in = U (5) : () 5 This denes an output frequency weighted (OFW) balanced truncation problem where the object to approximate is ~ N ~ M. The resulting ~ Nr and ~ Mr have the same state matrix and dene a LCF of ^Pr. Figure : Closed-loop conguration for the model P in feedback with the PID contrler PID. Consider a right coprime factorization (RCF) PID = U ; of the contrler. Some basic calculations show that We apply the OFW balanced truncation method to the stable normalized LCF of the system P. The reduced model of order is denoted ^P and the contrler of ^P order computed from this model is denoted. For the sake of further discussion and comparison, we have also computed the model ^P of order obtained by this procedure, and the corresponding contrler of p.

5 ^P order,. The approximation error has the same property as in open-loop reduction: kp ; ^P r k P i=r+ i (here, the i 's denote HS's of the OFW LCF). The same interpretation in the gap metric hds, since the same normalized LCF of P is used. 5 Second approach to a low-order contrler: contr design flowed by contrler reduction Our second approach consists in rst designing a highorder LQG contrler for the plant P and then reducing its order. The contr design criterion is as explained in Section. However the realization P is nearly non-minimal, which causes some numerical problems in the computation of the alman lter part of the contrler. Therefore 9 nearly cancelling modes must be dropped before designing the contrler, which consequently has order instead of 9. This optimal LQG contrler is denoted. 5. Open-loop coprime factor reduction being unstable, we have to perform the reduction on its normalized coprime factors. All the derivations of Subsection. are still valid here if we replace ~ N ~ M by (normalized) coprime factors ~U ~ of such that = ~ ; ~ U. In order to establish comparisons with the previoulsy computed low-order contrlers, we have reduced down to orders and, the corresponding contrlers being denoted ^ and ^. We have also computed the contrler of order 9 ^ 9 in order to compare it with ^ 9 which will be obtained in Subsection 5.. U Note that a right coprime factorization : = U ; can also be used, leading to the same results. 5. Closed-loop coprime factor reduction The basic idea that leads us to reduce the coprime factors of the contrler in osed-loop is the same as that used in Subsection. for osed-loop model reduction, where the osed-loop transfer function T (P PID ) was approximated by T ^Pr PID, P being reduced and PID remaining unchanged. Here, we approximate T (P ) by T P ^r, where the contrler is the object to be reduced and P is xed. Consider again equation (). We used it in Subsection. to transform the problem of reducing P in osed loop to that of reducing N ~ M ~ with frequency weightings depending on. Here we do U the opposite: the same equation can be used to transform the problem of reducing in osed loop to that U of reducing with weightings that now depend on ~N M ~ U. Of course, here, is no longer a U RCF of PID, but a normalized RCF of. ~N M ~ does no longer have to be normalized and can be chosen such that=i, which is assumed in the sequel. Minimizing the approximation error between the two osed-loop transfer functions T (P ) and T P ^r is thus equivalent with minimizing a frequency weighted dierence between the RCF's of and ^r : min ^ r T (P ) ; T P ^r m () min fu r rg W U Ur out ; W in Similarly to what was done in Subsection., the weightings are chosen in order to take account of the entries of y, r and d that are concerned by the contr objective, namely,, d Ohp, d Ucex and W ref. However, it seems very reasonable to require that the reduced-order contrler be as much as possible optimal with respect to the criterion () that was used to compute. Therefore, contrary to the choice made in Subsection., W out also penalizes the contr signals in u, namely and,andintegrates the contr criterion via weightings put on,, and, and selected to reect as much as possible the weightings imposed in the design criterion (). Consequently, the input and output frequency weighting lters are respectively and W out = W in = N ~ M ~ r 5 F (s) F (s) (8) 5 : (9) p. 5

6 Here the i 's are the square roots of the entries of Q in (), the i 's are those of R in (), and F (s) isthe LQG lter. Since F (s) = (s +)=s is unstable, we have replaced it by (s + ) = (s + ") where " = ;. This denes an input-output frequency weighted (IOFW) balanced truncation problem where the object U to approximate is. The resulting U r and r have the same state matrix and dene a RCF of ^r. In order to establish comparisons with the previoulsy computed low-order contrlers, we have reduced down to orders and, the corresponding contrlers being denoted ^ and ^. We have also computed the contrler of order 9, ^ 9,which is the lowest-order contrler that can be obtained by this method and that still has good performance when applied to P. Note that we have also tried to perform the reduction with a lter W out that only puts a unit penalty on and, and no penalty atallonthetwo contr signals. This case is thus the dual situation of Subsection.. The results proved much less satisfactory, with important oscillations appearing in the contr signals when the reduction was pushed down to order or less Figure : Responses to a step on W ref : P contrled by.5 ^P ( ), ^ (;;) and ^ x... (;) Comparative study of contrler performance In this section we compare the performance on the actual system P of the contrlers computed in the two previous sections Performance of contrlers of order We rst consider the four contrlers of order, ^P ^P, ^P, ^ and ^, obtained respectively via openloop model reduction, osed-loop model reduction, open-loop contrler reduction and osed-loop contrler reduction. destabilizes P. The other three havevery good performance when applied to P the last two are almost undistinguishable (their performance is very ose to that of, which is not illustrated in order to keep the plots readable) and a bit better than the second one when considering their responses to a step disturbance d Ucex, as shown in Figure. Figures and 5 show the osed-loop responses of P with those contrlers, resp. to a step reference W ref and to a step disturbance d Ohp.. Performance of contrlers of order Here we consider the three contrlers of order, ^P, ^ and ^, obtained respectively via osedloop model reduction, open-loop contrler reduction Figure 5: Responses to a step on d Ohp : P contrled by ^P ( ), ^ (;;) and ^ (;). and osed-loop contrler reduction. Recall that is the lowest order reachable by osed-loop model reduction while ensuring stability. Their performance are depicted in Figures to 9. Trends observed in the previous subsection are con- rmed here: the responses of, although still very acceptable, deviate from those of the other two contrlers, which remain ose to the optimal one (not shown on the plots).. Performance of contrlers of order 9 Finally, we consider the two contrlers of order 9, ^ 9 and ^ 9, obtained respectively via open-loop and osed-loop contrler reduction. Their performance are depicted in Figures to, where they are compared to that of the optimal LQG contrler. ^P p.

7 .5 x x Figure : Responses to a step on d Ucex: P contrled by ^P ( ), ^ (;;) and ^ (;). Figure 8: Responses to a step on d Ohp : P contrled by ^P ( ), ^ (;;) and ^ (;) Figure : Responses to a step on W ref : P contrled by ^P ( ), ^ (;;) and ^ (;). 5 5 Figure 9: Responses to a step on d Ucex: P contrled by ^P ( ), ^ (;;) and ^ Conusions (;). These plots early show the superiority of osed-loop contrler reduction when the order is reduced as much as possible. While the open-loop reduced contrler has unacceptable performance when applied to P,the osed-loop reduced one remains very ose to the optimal high-order contrler,even at such low order as 9. However, recall that such a good result has been obtained with weightings chosen to reect the design criterion (see Subsection 5.). Such weightings might be dicult to choose in some cases, while the weightings used in osed-loop model reduction are obtained in a much more starightforward way. In this study we have compared four dierent ways of obtaining a low-order contrler for a complex plant: open-loop or osed-loop reduction of the plant model flowed by contr design, and full-order contr design flowed by open-loop or osed-loop reduction of the optimal contrler. Furthermore, in [], a fth method was studied: direct osed-loop identication of a loworder plant model flowed by contr design. The methods have been tested on a realistic, high-order linearized model of a PWR nuear power plant, the goal being the replacement of two PID contrlers by a multivariable contrler for the electrical power while ensuring an acceptable water level regulation in the condenser. p.

8 .5. x Figure : Responses to a step on W ref : P contrled by ( ), ^ 9 (;;) and ^ 9 (;) Figure : Responses to a step on d Ucex: P contrled.. by ( ), ^ 9 (;;) and ^ 9 (;) Performance Method MAX CL contrler reduction # CL model identication CL model reduction ## OL contrler reduction MIN OL model reduction Figure : Responses to a step on d Ohp : P contrled by ( ), ^ 9 (;;) and ^ 9 (;). Flowing the observations made in Section of this paper and in [], two tables can be drawn up. Table assies the methods with respect to the performance achieved bycontrlers of the same order (when this order is suciently low to make dierences appear). The performance of each contrler is evaluated in terms of the discrepancy between its responses and those of the optimal one ( ). On the other hand, Table assies the methods with respect to the lowest order that can be reached with a pre-specied level of performance. Methods based on contrler reduction appear to be potentially more powerful than those based on model reduction, when they are both considered either in open loop or in osed loop. This result is quite logical since reduction implies a loss of information: it is best to keep all the information (i.e. the richness of the model) as long as possible. In other words, it may be dicult Table : Classication of the methods w.r.t. the performance achievable with contrlers of the same order to predict how model reduction will aect the to-bedesigned contrler while, on the other hand, when doing contrler reduction, the starting point is the optimal high-order contrler and the loss of performance is directly related to the approximation error committed during the reduction. However, contrler reduction has an important drawback: to give the best results, it requires specic weightings that are not completely dened by the objective ofmaintening the osed-loop transfer function, as it is the case in osed-loop model reduction. The weightings have tobechosen such that the reduction criterion matches as much as possible the design one, and this requirement might be dicult Order Method MIN CL contrler reduction " CL model reduction OL contrler reduction "" CL model identication MAX OL model reduction Table : Classication of the methods w.r.t. the order reachable with a pre-specied level of performance p. 8

9 to fulll in some cases. One might also argue that balanced truncation is possibly not the most appropriate method to reduce a contrler, since it rests on contrlability and observability notions that are more adapted to design models than to contrlers. In order to improve the quality of the results obtained via model reduction, further study will investigate the possible advantage of a matching between the contr design and the model reduction criteria (here, this matching was only considered in the contrler reduction approach, while model reduction was simply based on a osed-loop transfer function preservation criterion). We shall also consider using the full-order contrler (instead of the suboptimal PID contrler) in the frequency weightings used in model reduction. References [] P. Bendotti, B. Codrons, C. Falinower, and M. Gevers, \Contr-oriented low-order modelling of a complex PWR plant: a comparison between open-loop and osedloop techniques," in Proc. of the th IEEE Conf. on Decision and Contr, (Tampa, Florida), pp. 9{95, 998. [] D. Meyer, \A fractional approach to model reduction," in Proc. of the 988 American Contr Conf., pp. {, 988. [] T. Georgiou and M. Smith, \Optimal robustness in the gap metric," IEEE Trans. on Automatic Contr, v. AC-5, pp. {8, 99. [] M. idyasagar, \The graph metric for unstable plants and robustness estimates for feedback stability," IEEE Trans. on Automatic Contr, v. AC-9, pp. {, 98. [5] M. Gevers, Towards a Joint Design of Identication and Contr? Essays on Contr: Perspectives in the Theory and its Applications, pages -5, New York: Birkhauser, 99. [] M. Gevers, \Identication for contr," in Proc. of the 5th IFAC Symposium on Adaptive Contr and Signal Processing, (Budapest, Hungary), pp. {, 995. [] P. an den Hof and R. Schrama, \Identication and contr { osed-loop issues," Automatica, v., pp. 5{, December 995. [8] P. Bendotti and B. Bodenheimer, \Identication and H contr design for a Pressurized Water Reactor," in Proc. of the th IEEE Conf. on Decision and Contr, (Orlando, Florida), pp. {, 99. [9] L. Ljung, \Identication, model validation and contr," th IEEE Conf. on Decision and Contr, 99. Plenary lecture. [] C. Beck and P. Bendotti, \Model reduction for unstable uncertain systems," in Proc. of the th IEEE Conf. on Decision and Contr, (San Diego, California), pp. 98{, 99. This study has also conrmed an important nding previously made in [], which now early extends to contrler reduction. This nding is osed-loop techniques are always preferable to open-loop ones, since they allow to achieve a specied level of performance with lower-order contrlers alternatively, for a given order, contrlers obtained via open-loop techniques generally perform worse than those obtained via osed-loop methods. p. 9