Performance assessment for the water level control system in steam generator of the nuclear power...

Annals of Nuclear Energy 45 (2012) 94–105

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Annals of Nuclear Energy

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Performance assessment for the water level control system in steam generatorof the nuclear power plant

Zhi Zhang, Li-Sheng Hu ⇑Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China

a r t i c l e i n f o

Article history:Received 30 January 2011Received in revised form 13 February 2012Accepted 17 February 2012Available online 1 April 2012

Keywords:Performance assessmentTwo PI controller systemU-tube steam generatorWater level control system

0306-4549/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.anucene.2012.02.012

⇑ Corresponding author. Tel.: +86 021 3420 4543; fE-mail address: [email protected] (L.-S. Hu).

a b s t r a c t

The steam generator water level control system is the most important components of a nuclear powerplant. The operating steam generator water level control system is increasingly recognized as a capitalasset that should be routinely maintained and monitored. However, the control loop performance assess-ment is still an open problem; thus, the performance assessment technology will be brought into thesteam generator water level control system in nuclear power plants. Performance assessment methodsfor the plant with stable and unstable zeros of two PI controller systems are developed at all specificpower levels. The numerical examples will demonstrate the effectiveness of the proposed method.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

With the increasing demand for energy, the development of nu-clear energy, considered as one of the best cleanest forms of en-ergy, has received increasing attention worldwide, especially inChina. The nuclear power plant (NPP) generates electricity by driv-ing the armature coupled to a steam turbine. Steam is generated bythe u-tube steam generator (UTSG), whose water level should becontrolled in safe limits in order to maintain plant availabilityand economic feasibility of a NPP. Hence, a suitable controller ofSG water level control systems is the crucial for NPPs. In the pastthree decades, many excellent designs of the level controller havebeen reported. Irving et al. (1980) developed a simplified dynamicmodel, in which the model parameters change as the operationpoint changes. Many notable studies have been developed basedon this model (Cho and No, 1996; Kothare et al., 2000; Na, 2001,2003; Munasinghe et al., 2005; Ahmad, 2010). Although a varietyof control design techniques (e.g., PID, adaptive, model predictive,robust H1, fuzzy logic, etc.) were considered in the design of thelevel controller, fewer techniques exist for objective measures ofwater level control loop performance or, conversely, measures ofthe level of difficulty in controlling the water level from routineoperating data.

Unlike the controller design problem which focuses on develop-ing the control strategy, the controller performance assessmentproblem is concerned about whether the designed controller per-formance is in accordance with the required performance in the

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design stage. Some reports on the performance of control loopsindicate that a majority of these loops perform poorly in processindustries (Desborough, 2003; Desborough and Miller, 2001),due, apparently, to complex operating circumstances that deviatefrom those envisaged at the design stage. Many factors can contrib-ute to their performance deterioration, including sensor or actua-tor failure, equipment fouling, feedstock variability, productchanges, and seasonal influences (Harris et al., 1999). Hence, it isnecessary to offer control engineers a well-suited tool for deter-mining whether specified performance targets and response char-acteristics are being met by the controlled process variables (Jelali,2006). In the past two decades, research works on control loopsperformance assessment have attracted growing interest.

Consider there is a control system in operation for a process andwe are interested in the controller health and how well the con-troller has been performing. It is not necessary to perturb the pro-cess with extraneous test signals, because any test will inevitablydisrupt normal process operations. To test the performance of acontroller, only routine closed-loop operating data and some a pri-ori process knowledge can be obtained. Assessment of the currentcontroller may include (1) determination of the capability of therunning control system, (2) selection and design of a benchmarkfor performance assessment, (3) assessment and detection of poorperforming loops, (4) diagnosis of the underlying causes and sug-gestion of improvement measures (Harris et al., 1999; Jelali, 2006).

The key step of performance assessment is estimating the per-formance benchmark from the data, which is used to test the devi-ation from the current control performance. In other words, theassociated benchmark controller is comparable to the current con-troller. By virtue of valuable contributions from Harris (1989), time

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Fig. 1. Simplest schematic of a nuclear power plant.

Z. Zhang, L.-S. Hu / Annals of Nuclear Energy 45 (2012) 94–105 95

series analysis techniques could be used to find a suitable expres-sion for the feedback controller-invariant term from routine oper-ating data, as well as the subsequent use of this as a benchmark toassess control loop performance. The associated minimum vari-ance (MV) controller as benchmark controller can also be obtained,which can be traced to the work of Åström (1970). Thus, the MVperformance benchmark has become the basic and popular bench-mark. Subsequently, Huang and Shah (1999) extended the estima-tion of MV benchmark to the MIMO system by introducing theinteractor matrix and proposed the filtering and correlation (FCOR)analysis method. Some other excellent works on control loops per-formance assessment can also be found in reviews (Harris et al.,1999; Jelali, 2006; Qin, 1998), and the references therein. The sur-vey reveals a remarkable number of application case studies inrefining, petrochemical, chemical sectors, pulp & paper plants,and metal processing industries. However, no performance assess-ment applications appeared in the SG water level control system ofthe nuclear power plant.

The SG water level control system is the most important com-ponents of a NPP. Several studies (Menon and Parlos, 1992; Parryet al., 1985) show that the reactor trips are primarily attributedto SG control problems. Hence, the operating SG water level controlsystem is increasingly recognized as a capital asset that should beroutinely maintained and monitored. Although some excellentworks about monitoring the feedwater flow rate (Na et al., 2005;Yang et al., 2009; Lim et al., 2010), the sensor health (Na et al.,2006), and the loss of coolant accidents (Na et al., 2008), have beenreported recently, the problem about that how well the operatingcontroller has been performing is still open in this field.

In this paper, we will focus on the controller health and devel-oping the controller performance assessment method for a class ofSG water level control systems, namely, two PI controller systems.The performance assessment technology will be brought into anew industrial area, SG water level control in NPPs, where littlework has been done previously. However, the SG water level sys-tem have many characteristics that challenge the developmentand application of performance assessment techniques such as(1) nonlinear plant characteristics, (2) nonminimum phase plantcharacteristics, (3) sensor measurements, (4) constraints (Kothareet al., 2000). Considering the above challenges, this paper mayfacilitate the maturity of the performance assessment technologyfor SG water level control.

The remainder of this paper is organized as follows. Section 2presents the plant description and the considered SG water levelcontrol system, wherein MV performance benchmark and perfor-mance index are described. In terms of two PI controller systems,performance assessment methods for the plant with stable andunstable zeros are developed in Section 3. Section 4 presentsnumerical examples that show the effectiveness of the proposedmethod. Conclusions are stated in Section 5.

Table 1UTSG model parameters.

p(%)

g1 g2 g3 g4 s1 s2 s3 s Qs0

5 0.058 7.704 9.63 0.181 41.9 38.72 48.4 119.6 57.415 0.058 3.568 4.45 0.226 26.3 17.20 21.5 60.5 180.830 0.058 1.464 1.83 0.310 3.6 43.40 4.5 17.7 381.750 0.058 0.840 1.05 0.215 34.8 2.88 3.6 14.2 660.0

100 0.058 0.376 0.47 0.105 28.6 2.72 3.4 11.7 1435.0

2. Problem formulation and preliminaries

2.1. Process and control systems description

The nuclear reactor under consideration is a pressurized waterreactor (PWR). The simplest formula of the overall nuclear powerplant is shown in Fig. 1. As the most important components ofNPP, UTSG transfers thermal energy from the reactor coolant sys-tem in the primary side to the secondary side, thus generatingsteam. The generated steam of the secondary circuit is sent tothe turbine, which is coupled to an armature to generate electric-ity. To avoid plant shutdowns and system unavailability, the UTSGwater level should be regulated. The UTSG dynamics has highnonlinearity and nonminimum phase behavior. It is generally

approximated by the linearized model for a given power level. Inthis paper, the following model is considered (Kim, 1990; Muna-singhe et al., 2005):

YðsÞ ¼ g1

s½Q wðsÞ � Q sðsÞ� �

g2

1þ s2sQwðsÞ þ

g3

1þ s3sQ sðsÞ

þ g4ss�2

2 þ 4p2s�2 þ 2s�11 sþ s2

Q wðsÞ ð1Þ

where s is the Laplace variable, s1, s2 and s3 are the dampingtime constants, s is the period of the mechanical oscillation, g1 isthe magnitude of the mass capacity effect, g2 is the magnitude ofthe swell or shrink due to the feedwater, g3 is the magnitudeof the swell or shrink due to steam flow rates, g4 is the magnitudeof the mechanical oscillation. The four terms on the right-handside are mass capacity effect, nonminimum phase effect of feedwater, nonminimum phase effects of steam, and effect of mechani-cal oscillation. The transfer function relates to the feed water flowrate, Qw, and the steam flow rate, Qs, to the water level, Y. The modelparameters of (1) are given in Table 1. Water level responses to thestep increase in feed water and steam flow rates at different oper-ating powers using the considered model are shown in Fig. 2.

.There are many successful control structures for regulating thewater level at all specific power levels, as given in Table 1. The twoPI controller structure is considered (Munasinghe et al., 2005), onefor level error control and another one for the flow error control.The structure of the two PI controllers is shown in Fig. 3. The level

error PI controller is GC1 ¼kl

psþkli

s , and the flow error PI controller is

GC2 ¼kf

psþkfi

s . The PI parameters are shown in Table 2 (Munasingheet al., 2005).

2.2. Minimum variance performance benchmark

The performance of a control system relates to its ability to dealwith the deviations between controlled variables and their set-points. The most popular stochastic performance criterion is the

0 50 100 150 200−4

−2

0

2

4

6

8

10

12

5%

15%

30%

50%

100%

Time (Sec)

Wat

er le

vel (

mm

)

(a)

0 50 100 150 200−12

−10

−8

−6

−4

−2

0

2

4

5%15%

30%50%

Time (Sec)

Wat

er le

vel (

mm

)

(b)

100%

Fig. 2. Response of the water level at different operating power levels (indicated by %) to (a) a step in feed water flow rate and (b) a step in steam flow rate.

Fig. 3. Two PI controller system to regulate UTSG water level.

Table 2PI gains for specific power levels.

p (%) klp kl

i kfp kf

i

5 2.70 0.09 0.50 1.0015 2.80 0.20 0.50 1.0030 1.50 0.70 0.50 1.0050 2.00 0.70 0.50 1.00

100 2.80 2.00 0.50 1.00

96 Z. Zhang, L.-S. Hu / Annals of Nuclear Energy 45 (2012) 94–105

variance, owing to its direct relationship to product quality andmaterial/energy consumption parameters. When MV of the systemoutput is selected as the benchmark, performance assessment willtest the deviation of the current control performance from thisbenchmark, and thus determine the possible improvement byenhancing the control performance from the current one to thatof the benchmark (Jelali, 2006).

Consider a SISO process under regulatory control as shown inFig. 4,

yt ¼ q�dGðq�1Þut þ Nðq�1Þat ; ð2Þ

Fig. 4. Block diagram of feedback system.

where the set-point is zero. yt is the variation of the output aroundthe set-point, ut is the control action, at is a Gaussian white noisewith zero mean and variance r2. q�d is a d-step time delay, d P 1in the control channel, and q�1 is the backward shift operator.G(q�1) is the delay-free plant transfer function, N(q�1) is the distur-bance transfer function, and ut = �Q(q�1)yt is the feedback controllaw. To simplify the sequel development, q�1 will be omitted unlesscircumstances necessitate its presence.

As discussed in (Harris, 1989; Huang and Shah, 1999), we havethe Diophantine identity as follows,

N ¼ f0 þ f1q�1 þ � � � þ fd�1q�dþ1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}F

þRq�d; ð3Þ

where fi, for i = 1, . . . ,d � 1, are impulse response coefficients of thedisturbance transfer function, and R is the remaining rational, prop-er transfer function. It can split the polynomial N into two terms,one related to the past values of at and the other related to the fu-ture values of at. Substituting (3) into (2) gives

yt ¼F þ q�1R

1þ q�dGQat

¼ F þ R� FGQ1þ q�dGQ

q�d

� �at

¼ Fat þ Lat�d;

ð4Þ

where L ¼ R�FGQ1þq�dGQ is a proper transfer function. The two terms on the

right-hand side of (4) are independent. Therefore, the variance ofthe output can be obtained as:

VarðytÞ ¼ VarðFatÞ þ VarðLat�dÞ:

Note that Var(�) denotes the variance operator. Hence, the varianceof the output satisfies the following inequality

VarðytÞP VarðFatÞ:

The equality holds when L = 0, i.e., R � FGQ = 0. It yields the MV con-trol law: Q ¼ R

GF. Hence, ytjmv = Fat is the process output under min-imum variance control independent to the controller action, andVar(ytjmv) is considered as MV performance benchmark, which istheoretically the lower bound of the output variance. If a stableclosed-loop output data yt is modeled by an infinite moving-averagemodel, then the MV performance benchmark can be estimated bythe first d terms. In terms of performance assessment, the deviationof the current controller performance from the MV benchmark canbe obtained (Huang and Shah, 1999).


Remark 1. The adoption of MV controller as a benchmark does notimply that it should be the goal toward which the existing controlshould be driven. However, implementation of MV control is notrecommended in practice owing to its poor robustness andexcessive control action. Nevertheless, the MV control sets aperformance bound for all other controllers in terms of the outputvariance. Hence, it serves as an appropriate benchmark againstwhich the performance of other controllers may be compared(Harris et al., 1999; Jelali, 2006).

The deviation can be quantified by the performance index.Harris (1989) suggested the controller performance index based onMV control. The performance index is defined as follows:

gðdÞ ¼ JmvJ¼ VarðytjmvÞ

VarðytÞ¼Pd�1

i¼0 f 2i ðtÞP1

i¼0f 2i ðtÞ

ð5Þ

The advantage of this performance index is that MV benchmark canbe calculated from routine operating data by estimating the coeffi-cients fi of the impulse response from noise-to-output transfer func-tion. This definition of the controller performance index satisfies0 6 g(d) 6 1, and provides a bounded and normalized performanceindicator. The distance between minimum variance control systemand actual control system can be directly seen. The value g(d) = 1indicates an ideal case of minimum variance control, whereasg(d) = 0 shows the case of the worst control.

Based on the above description, the performance assessmentmethod will be developed for a class of SG water level controlsystems, two PI controller systems, in the next section.

3. Performance assessment for the SG water level controlsystem

3.1. MV performance benchmark of two PI controller systems withoutunstable zeros in plants

In this section, the MV performance benchmark of the two PIcontroller loops for regulating UTSG water level is obtained whenstochastic load disturbances occur in feed water flow rate andwater level. Fig. 5 shows a discretized two PI controller system.For the water level, feed water flow rate, and output of the secondPI controller GC2, we have GL1 = GdGs, and

y1t ¼ GL1a1t þ G1y2t ð6Þy2t ¼ G2ðu2t � GC1y1tÞ þ GL2a2t ð7Þu2t ¼ GC2ðGda1t � y2tÞ ð8Þ

where y1t is the deviation of water level output from its set-point,and y2t is the deviation of feed water flow rate output from its stea-dy state value. G1 denotes the transfer function from feed waterflow rate to water level, where G1 ¼ G1q�d1 ; q�d1 is a d1-step timedelay, G1 is the delay-free transfer function, assuming all zeros of

Fig. 5. Two PI controller system.

G1 lie inside the unit circle. G2 denotes the transfer function fromthe sum of the output of the first PI controller and the second PIcontroller to feed water flow rate, G2 ¼ G2q�d2 ; q�d2 is a d2-steptime delay, and G2 is the delay-free transfer function, assuming thatG2 is also the minimum phase. Gs denotes the transfer function fromsteam flow rate disturbance to water level. The disturbance transferfunctions, Gd and GL2 are assumed to be a rational function of q�1,and they are driven by the Gaussian white noise sequences a1t

and a2t with zero mean and variance r2a1

and r2a2

, respectively.From (6)–(8), we have

y1t ¼ð1þ G2GC2ÞGL1a1t þ G1G2GC2Gda1t þ G1GL2a2t

ð1þ G2GC2 þ G1G2GC1Þð9Þ

There are Diophantine identities as follows:

GL1 ¼ F1 þ R1q�d1�d2 ð10Þ

GL2 ¼ F2 þ R2q�d2 ð11Þ

G1F2 ¼ Sþ Tq�d2 ð12Þ

where F1 is a polynomial in q�1 of the order d1 + d2 � 1, and F2 and Sare polynomials in q�1 of the order d2 � 1, and R1, R2 and T are prop-er transfer functions.

Substituting (10)–(12) to (9), and consideringG1 ¼ G1q�d1 ; G2 ¼ G2q�d2 , we have

y1t ¼ð1þG2GC2ÞðF1þR1q�d1�d2 Þa1t þG1G2GC2Gda1t þG1ðF2þR2q�d2 Þa2t

1þG2GC2þG1G2GC1

¼ F1a1t þð1þG2GC2ÞR1a1tq�d1�d2 þG2GC2F1a1t þG1G2GC2Gda1t

1þG2GC2þG1G2GC1

þðSþ Tq�d2 Þq�d1 a2t þG1R2q�d1�d2 a2t

1þG2GC2þG1G2GC1

¼ F1a1t þ Sq�d1 a2t þq�d1�d2

� ð1þG2GC2ÞR1þG1G2GC2Gd� F1G1G2GC1

1þG2GC2þG1G2GC1

" #|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

L1

a1t þq�d1�d2

� TþG1R2� SG2ðGC2þG1GC1Þ1þG2GC2þG1G2GC1

" #|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

L2

a2t

¼ F1a1t þ Sq�d1 a2t|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}controller�independent

þq�d1�d2 ½L1a1t þ L2a2t �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}controller�dependent

;

where L1 and L2 are proper. The controller-independent term isindependent to the control action and the controller-dependentterm. Hence, the variance of the water level output satisfies the fol-lowing inequality:

VarðytÞP VarðF1a1t þ Sq�d1 a2tÞ:

The equality holds when L1 = 0 and L2 = 0. We thus have the MV per-formance benchmark VarðytjmvÞ ¼ VarðF1a1t þ Sq�d1 a2tÞ, and

R1 þ R1G2GC2 þ G1G2GC2Gd � F1G1G2GC1 ¼ 0

T þ G1R2 � SG2GC2 � SG2G1GC1 ¼ 0:

Hence, the MV controllers can be obtained

GC1 ¼ðT þ G1R2ÞðGL1 þ G1GdÞ � ðF1T þ F1G1R2 � SR1q�d1 Þ

SG2G1ðGL1 þ G1GdÞ

GC2 ¼F1T þ F1G1R2 � SR1q�d1

SG2ðGL1 þ G1GdÞ:


3.2. MV performance benchmark of two PI controller systems withunstable zeros in plants

Commonly, UTSG plants may have nonminimum phase dynam-ics. When processes have unstable zeros, controllers may not bephysically realizable, and may not be stable and proper (Ko andEdgar, 2000; Tyler and Morari, 1995). In this subsection, we con-sider the process G1 to have unstable zeros and G2 does not haveunstable zeros. We have G1 ¼ Gþ1 G�1 q�d1 , where Gþ1 is a monic poly-nomial that contains all the unstable zeros of G1, and G�1 q�d1 is theremaining term of G1. There are Diophantine identities as follows:

GL1 ¼ F1 þ R1q�d1�d2 ð13ÞGL2 ¼ F2 þ R2q�d2 ð14ÞeG1F2 ¼ eS þ eT q�d2 ; ð15Þ

where eG1 ¼ q�nGþ1 ðqÞG�1 ðq�1Þ; n ¼ degðGþ1 Þ; F1 is a polynomial in

q�1 of the order d1 + d2 � 1, and F2 and eS are polynomials in q�1

of the order d2 � 1, and R1, R2 and eT are proper transfer functions.Note that deg(�) denotes the degree of a polynomial.

Substituting (13)–(15) to (9), and considering D(q�1)D(q) = 1,DG1 ¼ eG1q�d1 , and G2 ¼ G2q�d2 , we have

Dy1t ¼DF1ð1þ G2GC2Þa1t þ DR1ð1þ G2GC2Þq�d1�d2 a1t

ð1þ G2GC2 þ G1G2GC1Þ

þeG1G2GC2Gdq�d1 a1t þ eG1F2q�d1 a2t þ eG1R2q�d1�d2 a2t

ð1þ G2GC2 þ G1G2GC1Þð16Þ

Consider the following identity

DR1 ¼ Rmp1 þ Rnmp

1 ;

where Rnmp1 is the term containing all the unstable zeros of G1 as its

poles after the partial fraction expansion of DR1. Rmp1 is the remain-

ing term after the partial fraction expansions. Thus (16) can be writ-ten as

Dy1t ¼ DF1a1t þ Rnmp1 q�d1�d2 a1t þ eSq�d1 a2t þ q�d1�d2

� Rmp1 ð1þ G2GC2Þ þ eG1G2GC2Gd � DF1G1G2GC1

1þ G2GC2 þ G1G2GC1

" #|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

L1

a1t

þ q�d1�d2eT þ eG1R2 � eSG2ðGC2 þ G1GC1Þ

1þ G2GC2 þ G1G2GC1

" #|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

L2

a2t

¼ DF1a1t þ q�d1�d2 Rnmp1 a1t þ eSq�d1 a2t|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

controller�independent

þ q�d1�d2 ½L1a1t þ L2a2t�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}controller�dependent

;

ð17Þ

where L1 and L2 are proper. The controller-independent term isindependent to the control action and the controller-dependentterm.

Remark 2. Consider the operator 1/(1 + aq�1) where jaj > 1(Åström and Wittenmark, 1990; Wiener, 1949). This operator isnormally interpreted as a causal unstable operator. Since jaj > 1and the shift operator has the norm jqj = 1, the series expansion

11þ aq�1 ¼

qa

11þ q=a

¼ qa

1� 1a

qþ 1a2 q2 � � � �

� �

converges. Thus, the operator 1/(1 + aq�1) can be interpreted as anoncausal stable operator. Therefore, q�d1�d2 Rnmp

1 in (17) can beexpanded in terms of the q operator. Assuming that Rnmp

1 ¼=ð1þ aq�1Þ, the expansion

q�d1�d2

1þ aq�1 ¼1a

q�d1�d2þ1

1þ q=a

¼ 1a

q�d1�d2þ1 � 1a

q�d1�d2þ2 þ 1a2 q�d1�d2þ3 � � � �

� �converges. The most recent term in this expansion is 1

a q�d1�d2þ1

which cannot be controlled by the control action. Hence, the termq�d1�d2 Rnmp

1 can be considered as the controller-independent (Huangand Shah, 1999).

Since D(q�1)D(q) = 1, yt and Dyt has the same variance, thevariance of the water level output satisfies the followinginequality:

VarðytÞ ¼ VarðDytÞP VarðDF1a1t þ Rnmp1 q�d1�d2 a1t þ eSq�d1 a2tÞ:

The equality holds when L1 ¼ 0 and L2 ¼ 0, where we have the MV

performance benchmark VarðytjmvÞ ¼ Var F1a1t þRnmp

1D q�d1�d2 a1tþ

�eSD q�d1 a2t

�, and

Rmp1 þ Rmp

1 G2GC2 þ eG1G2GC2Gd � DF1G1G2GC1 ¼ 0eT þ G1R2 � eSG2GC2 � eSG2G1GC1 ¼ 0

Hence, the MV controllers can be obtained

GC1 ¼ðeT þ eG1R2ÞðRmp

1 q�d1�d2 þ eG1Gdq�d1 Þ þ eSRmp1 q�d1eSG2G1ðDF1 þ Rmp

1 q�d1�d2 þ eG1Gdq�d1 Þ

GC2 ¼DF1

eT þ DF1eG1R2 � eSRmp

1 q�d1

DF1eSG2 þ eSðRmp

1 G2 þ eG1G2GdÞq�d1

3.3. Estimation of minimum variance performance benchmark

The MV performance benchmark can be estimated from routineoperating data with some a priori knowledge.

3.3.1. The plant without unstable zerosFor the plant without unstable zeros, a multivariate time series

model of y1t can be obtained with a priori knowledge of the processtime delays. It can be described as the closed-loop transfer func-tions that relate [a1t a2t] to y1t. The first d1 + d2 moving averagecoefficients are all controller-independent terms,

y1tjmv ¼ bF 1a1t þ bSq�d1 a2t ; ð18Þ

where bF 1 is an estimated polynomial in q�1 of the order d1 + d2 � 1,and bS is an estimated polynomial in q�1 of the order d2 � 1. Theinnovations sequences a1t and a2t are estimated as the residual vec-tors of multivariate multiple linear regression analysis. Thus, theMV performance benchmark is estimated bybJmv ¼ VarðbF 1a1t þ bSq�d1 a2tÞ

¼ trace ðXd1þd2�1

i¼0

bNTibNiÞ � bRa

" #;

ð19Þ

where bNi, for i = 0, . . . ,d1 + d2 � 1, is the coefficient matrix of theestimated matrix polynomial. ½bF 1

bSq�d1 �, and bRa is the estimatedcovariance matrix of the white noise vector ½a1t a2t �T . Note thattrace(�) denotes the trace of a square matrix.

3.3.2. The plant with unstable zerosFor the plant with unstable zeros, we can also obtain a multivar-

iate time series model of y1t with a priori knowledge of the processtime delays, and the closed-loop transfer function from [a1t a2t] toy1t,

y1t ¼ bGN1a1t þ bGN2a2t : ð20Þ


Multiply (20) by D,

Dy1t ¼ DbGN1a1t þ DbGN2a2t ; ð21Þ

and (21) can be expanded as

Dy1t ¼ ðDbH1 þ DbI1q�d1�d2 Þa1t þ ðDbH2 þ DbI2q�d1�d2 Þa2t ; ð22Þ

where bH1 ¼ bF 1; bH2 ¼^eSD q�d1 , are estimated polynomials in q�1 of the

order d1 + d2 � 1, and the term bRnmp1 can be estimated from the

transfer functions that have all unstable zeros of the G1 as their

poles after the partial fraction expansion of DbI1 with a priori knowl-edge of the locations of unstable zeros (Huang and Shah, 1999).

Thus, MV performance benchmark is estimated by

bJmv ¼ Var bF 1 þbRnmp

1

Dq�d1�d2

!a1t þ

eSD

q�d1 a2t

0@ 1A¼ trace ð

Xd1þd2�1

i¼0

bNTibNiÞ � bRa

" #;

ð23Þ

where bNi, for i = 0, . . . ,d1 + d2 � 1, is the coefficient matrices of the

estimated matrix polynomial bF 1 þbRnmp

1D q�d1�d2

� � beSD q�d1

" #, and bRa

is the estimated covariance matrix of the white noise vector½a1t a2t�T .

4. Numerical example

Consider the SG water level system controlled by two PI con-trollers. The control structure is shown in Figs. 3 and 5. Supposethat the system at each power level has been well modeled. Themodel parameters of UTSG are listed in Table 1. The system sam-pling interval is 1 s, and there is a 2 s pure time delay in processG1(q�1). a1t and a2t are Gaussian white noise sequences with zeromean and variance 0.05 and 0.04, respectively.

(1) When the plant operates at 5% of its rated power, we can ob-tain that the following:

G1 ¼0:0383ð1� 0:852q�1Þð1� 2:1714q�1 þ 1:189q�2Þq�2

1� 3:9245q�1 þ 5:7782q�2 � 3:7828q�3 þ 0:9291q�4 ; d1

¼ 2;

G2 ¼q�1

1� 0:1q�1 ; d2 ¼ 1;

Gs ¼0:1389q�1 � 0:1401q�2

1� 1:9796q�1 þ 0:9796q�2 ; Gd ¼1

1� 0:1q�1 ;

GL1 ¼ Gd � Gs

¼ ð0:1389q�1 þ 0:1488q�2Þ

þ 0:1458� 0:1616q�1 þ 0:0146q�2

1� 2:0796q�1 þ 1:1776q�2 � 0:098q�3 q�3;

F1 ¼ 0:1389q�1 þ 0:1488q�2; R1

¼ 0:1458� 0:1616q�1 þ 0:0146q�2

1� 2:0796q�1 þ 1:1776q�2 � 0:098q�3 ;

GL2 ¼1

1� 0:2q�1 ¼ 1þ 0:21� 0:2q�1 q�1; F2 ¼ 1;

R2 ¼0:2

1� 0:2q�1 :

Since G1(q�1) has two unstable zeros, 1.0857 + 0.1011i and 1.0857–0.1011i, we have

Gþ1 ðq�1Þ ¼ 1� 2:1714q�1 þ 1:189q�2;

eG1F2 ¼ q�nGþ1 ðqÞG�1 ðq�1ÞF2

¼ q�2ð1� 2:1714q�1 þ 1:189q�2Þ

� 0:0383ð1� 0:852q�1Þ1� 3:9245q�1 þ 5:7782q�2 � 3:7828q�3 þ 0:9291q�4

¼ 0:0455

þ 0:0566� 0:1537q�1 þ 0:1395q�2 � 0:0423q�3

1� 3:9245q�1 þ 5:7782q�2 � 3:7828q�3 þ 0:9291q�4 q�1;

eS ¼ 0:0455; eT¼ 0:0566� 0:1537q�1 þ 0:1395q�2 � 0:0423q�3

1� 3:9245q�1 þ 5:7782q�2 � 3:7828q�3 þ 0:9291q�4 ;

and

D ¼ q�nGþ1 ðqÞ=Gþ1 ðq�1Þ ¼ q�2ð1� 2:1714qþ 1:189q2Þ1� 2:1714q�1 þ 1:189q�2

Thus,

DR1 ¼q�2ð1� 2:1714qþ 1:189q2Þ1� 2:1714q�1 þ 1:189q�2

� 0:1458� 0:1616q�1 þ 0:0146q�2

1� 2:0796q�1 þ 1:1776q�2 � 0:098q�3

¼ 0:011þ 0:1589q�1

1� 2:1714q�1 þ 1:189q�2

þ 0:0145� 0:0159q�1 � 0:1176q�2

1� 2:0796q�1 þ 1:1776q�2 � 0:098q�3 ;

Rnmp1 ¼ 0:011þ 0:1589q�1

1� 2:1714q�1 þ 1:189q�2 ; Rmp1

¼ 0:0145� 0:0159q�1 � 0:1176q�2

1� 2:0796q�1 þ 1:1776q�2 � 0:098q�3 :

Hence, the MV performance benchmark can be obtained

yt jmv ¼ F1a1t þRnmp

1

Da1tq�d1�d2 þ

eSD

a2tq�d1

¼ ½0:1389q�1 þ 0:1488q�2 þ 0:011þ 0:1589q�1

1� 2:1714q�1 þ 1:189q�2

� 1� 2:1714q�1 þ 1:189q�2

1:189� 2:1714q�1 þ q�2 q�3�a1t þ 0:0455

� 1� 2:1714q�1 þ 1:189q�2

1:189� 2:1714q�1 þ q�2 q�2a2t :

Considering Remark 2, this can be simplified as

yt jmv ¼ ð0:1389q�1 þ 0:1488q�2 þ 0:0093q�3Þa1t þ ð0:0383q�2Þa2t :

Hence, Jmv = Var(ytjmv) = 0.0021.For the sake of illustration, similar to Huang and Shah (1999),

consider the GC1 as the form GC1 ¼ k12:7�2:61q�1

1�q�1 , and the GC2 as the

form GC2 ¼ 0:5þ0:5q�1

1�q�1 . Applying the proposed assessment method

yields the results shown in Fig. 6a. For this form of control, themaximum performance measure is about 0.2153 when the variablek1 takes the value of about 1.65. There may be a possible improve-ment by enhancing the control performance from the current oneto that of the MV performance benchmark. Similarly, we consider

0 0.5 1 1.5 2 2.40

0.05

0.1

0.15

0.2

0.25

0.3

Variable k1

(a)

0 0.5 1 1.5 2 2.40

0.05

0.1

0.15

0.2

0.25

0.3

Variable k2

Perfo

rman

ce in

dex

(η)

(b)

Perfo

rman

ce in

dex

(η)

Fig. 6. Control loop performance measure for k1 and k2 at 5% power level, respectively.


the GC1 as the form GC1 ¼ 2:7�2:61q�1

1�q�1 , and the GC2 as the form

GC2 ¼ k20:5þ0:5q�1

1�q�1 . Applying the proposed assessment method yields

the results shown in Fig. 6b. The maximum performance measureis about 0.2206 when the variable k2 = 0.6. There may also be thepossible improvement by enhancing the control performance.

Furthermore, consider the form of GC1 is k12:7�2:61q�1

1�q�1 , and the

form of GC2 is k20:5þ0:5q�1

1�q�1 , the performance measure can be de-

scribed as Fig. 7a and the contour is also obtained as shown inFig. 7b. The maximum performance measure is about 0.2; thusthe control performance can be possibly improved.

(2) When the plant operates at 15% of its rated power, we canobtain that the following:

G1 ¼ ð1� 8:0906q�1Þ

� 0:0736ð1� 0:3434q�1 þ 0:1156q�2Þq�2

1� 3:8565q�1 þ 5:5882q�2 � 3:6062q�3 þ 0:8744q�4 ;

G2 ¼q�1

1� 0:1q�1 ; d1 ¼ 2; d2 ¼ 1;

Gs ¼0:1442q�1 � 0:1468q�2

1� 1:954q�1 þ 0:9546q�2 ; Gd ¼1

1� 0:1q�1 ;

GL1 ¼ Gd � Gs

¼ ð0:1442q�1 þ 0:1495q�2Þ

þ 0:1414� 0:1581q�1 þ 0:0143q�2

1� 2:0546q�1 þ 1:1501q�2 � 0:0955q�3 q�3;

0 0.5 1 1.5 2 2.4

00.5

11.5

22.40

0.050.1

0.150.2

0.250.3

Variable k1

(a)

Variable k2

Perfo

rman

ce in

dex

(η)

Fig. 7. Control loop performance measure for

F1 ¼ 0:1442q�1 þ 0:1495q�2; R1

¼ 0:1414� 0:1581q�1 þ 0:0143q�2

1� 2:0546q�1 þ 1:1501q�2 � 0:0955q�3 ;

GL2 ¼1

1� 0:2q�1 ¼ 1þ 0:21� 0:2q�1 q�1; F2 ¼ 1; R2

¼ 0:21� 0:2q�1 :

Since G1(q�1) has an unstable zero, 8.0906, we have

Gþ1 ðq�1Þ ¼ 1� 8:0906q�1;


¼ q�1ð1�8:0906qÞ

� 0:0736ð1�0:3434q�1þ0:1156q�2Þ1�3:8565q�1þ5:5882q�2�3:6062q�3þ0:8744q�4

¼�0:5955

þ �2:0184þ3:2337q�1�2:139q�2þ0:5207q�3

1�3:8565q�1þ5:5882q�2�3:6062q�3þ0:8744q�4 q�1;

eS ¼ �0:5955; eT¼ �2:0184þ 3:2337q�1 � 2:139q�2 þ 0:5207q�3

1� 3:8565q�1 þ 5:5882q�2 � 3:6062q�3 þ 0:8744q�4 ;

0.1

0.1

0.10.1

0.2

0.2

0.2

0.2

Variable k1

Varia

ble

k 2

(b)

0 0.5 1 1.5 2 2.40

0.5

1

1.5

2

2.4

k1 and k2, and contour at 5% power level.


and

D ¼ q�nGþ1 ðqÞ=Gþ1 ðq�1Þ ¼ q�1ð1� 8:0906qÞ1� 8:0906q�1 ;

DR1 ¼q�1ð1� 8:0906qÞ

1� 8:0906q�1 � 0:1414� 0:1581q�1 þ 0:0143q�2

1� 2:0546q�1 þ 1:1501q�2 � 0:0955q�3

¼ 0:01331� 8:0906q�1 þ

�0:1456� 1:2739q�1 þ 0:1299q�2

1� 2:0546q�1 þ 1:1501q�2 � 0:0955q�3 ;

Rnmp1 ¼ 0:0133

1� 8:0906q�1 ; Rmp1

¼ �0:1456� 1:2739q�1 þ 0:1299q�2

1� 2:0546q�1 þ 1:1501q�2 � 0:0955q�3 :


ytjmv ¼ F1a1t þRnmp

1

Da1tq�d1�d2 þ

eSD

a2tq�d1

¼ ½0:1442q�1 þ 0:1495q�2 þ 0:01331� 8:0906q�1

� 1� 8:0906q�1

�8:0906þ q�1 q�3�a1t � 0:5955 � 1� 8:0906q�1

�8:0906þ q�1 q�2a2t;

this can be simplified as

ytjmv ¼ ð0:1442q�1 þ 0:1495q�2 � 0:0016q�3Þa1t þ ð0:0736q�2Þa2t:

Hence, Jmv = Var(ytjmv) = 0.0024.Similarly, consider GC1 ¼ k1

2:8�2:6q�1

1�q�1 , GC2 ¼ 0:5þ0:5q�1

1�q�1 , the perfor-

mance measure can be described as Fig. 8a. When k1 = 1.15, the

maximum value is 0.2772. Consider GC1 ¼ 2:8�2:6q�1

1�q�1 , GC2 ¼k2

0:5þ0:5q�1

1�q�1 , the performance measure can be described as Fig. 8b.When k2 = 0.7, the maximum value is 0.3028. Furthermore, consider

GC1 ¼ k12:8�2:6q�1

1�q�1 , GC2 ¼ k20:5þ0:5q�1

1�q�1 , the performance measure can be

described as Fig. 9a and the contour is also obtained as shown inFig. 9b. The maximum performance measure is about 0.3. Threecases show that the control performance can be possibly improved.


G1 ¼0:2576ð1� 0:9624q�1Þð1� 1:9614q�1 þ 0:9721q�2Þq�2

1� 3:4552q�1 þ 4:4732q�2 � 2:5786q�3 þ 0:5606q�4 ;

d1 ¼ 2;

G2 ¼q�1

1� 0:1q�1 ; d2 ¼ 1;

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Variable k1

(a)

Perfo

rman

ce in

dex

(η)

Fig. 8. Control loop performance measure for k

Gs ¼0:2887q�1 � 0:3183q�2

1� 1:8007q�1 þ 0:8007q�2 ; Gd ¼1

1� 0:1q�1 ;

GL1 ¼ GdGs

¼ ð0:2887q�1 þ 0:2304q�2Þ

þ 0:1547� 0:2029q�1 þ 0:0185q�2

1� 1:9007q�1 þ 0:9808q�2 � 0:0801q�3 q�3;

F1 ¼ 0:2887q�1 þ 0:2304q�2;

R1 ¼0:1547� 0:2029q�1 þ 0:0185q�2

1� 1:9007q�1 þ 0:9808q�2 � 0:0801q�3 ;

GL2 ¼1

1� 0:2q�1 ¼ 1þ 0:21� 0:2q�1 q�1; F2 ¼ 1;

R2 ¼0:2

1� 0:2q�1 :

Since G1 has no unstable zero, we have

G1F2 ¼ 0:2576

þ 0:1369� 0:4147q�1 þ 0:4232q�2 � 0:1444q�3

1� 3:4552q�1 þ 4:4732q�2 � 2:5786q�3 þ 0:5606q�4 q�1

S ¼ 0:2576; T

¼ 0:1369� 0:4147q�1 þ 0:4232q�2 � 0:1444q�3

1� 3:4552q�1 þ 4:4732q�2 � 2:5786q�3 þ 0:5606q�4 :


yt jmv ¼ F1a1t þ Sq�d1 a2t

¼ ð0:2887q�1 þ 0:2304q�2Þa1t þ 0:2576q�2a2t:


1:5�0:8q�1

1�q�1 ; GC2 ¼ 0:5þ0:5q�1

1�q�1 , Fig. 10a canbe obtained. When k1 = 0.15, the maximum performance measure

is 0.3694. Consider GC1 ¼ 1:5�0:8q�1

1�q�1 ; GC2 ¼ k20:5þ0:5q�1

1�q�1 , Fig. 10b can

be obtained. When k2 = 0.975, the maximum performance measure

is 0.3548. Consider GC1 ¼ k11:5�0:8q�1

1�q�1 ; GC2 ¼ k20:5þ0:5q�1


mance measure can be described as Fig. 11a and the contour is alsoobtained as shown in Fig. 11b. The maximum performance mea-sure is about 0.35. Three cases show that the control performancecan also be possibly improved.


0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Variable k2

Perfo

rman

ce in

dex

(η)

(b)

1 and k2 at 15% power level, respectively.

00.5

11.5

2

00.5

11.5

20

0.050.1

0.150.2

0.250.3

0.350.4

Variable k1

(a)

Variable k2

Perfo

rman

ce in

dex

(η)

0.15

0.15

0.15

0.15

0.3

0.3

Variable k1

Varia

ble

k 2

(b)

0 0.5 1 1.5 20

0.5

1

1.5

2

Fig. 9. Control loop performance measure for k1 and k2, and contour at 15% power level.

0 0.5 1 1.5 20

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

Variable k1

(a)

0 0.5 1 1.5 20

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

Variable k2

Perfo

rman

ce in

dex

(η)

(b)

Perfo

rman

ce in

dex

(η)


00.5

11.5

00.5

11.5

0

0.2

0.4

Variable k1

(a)

Variable k2

Perfo

rman

ce in

dex

(η)

0.05

0.05

0.05

0.05

0.1

0.1

0.1

0.1

0.15

0.15

0.15

0.2

0.2

0.2

0.25

0.25

0.25

0.3

0.3

0.3

0.35

0.35

0.35

Variable k1

Varia

ble

k 2

(b)

0 0.5 1 1.50

0.5

1

1.5



G1 ¼0:0097ð1� 2:8225q�1Þð1� 1:1057q�1Þð1þ 1:7839q�1Þq�2

1� 3:3516q�1 þ 4:4581q�2 � 2:7736q�3 þ 0:6671q�4 ;

d1 ¼ 2;

G2 ¼q�1

1� 0:1q�1 ; d2 ¼ 1;

Gs ¼0:1967q�1 � 0:2108q�2

1� 1:7575q�1 þ 0:7575q�2 ; Gd ¼1

1� 0:1q�1 ;

GL1 ¼ GdGs

¼ ð0:1967q�1 þ 0:1546q�2Þ

þ 0:0991� 0:1294q�1 þ 0:0117q�2

1� 1:8575q�1 þ 0:9333q�2 � 0:0757q�3 q�3;

F1 ¼ 0:1967q�1 þ 0:1546q�2; R1

¼ 0:0991� 0:1294q�1 þ 0:0117q�2

1� 1:8575q�1 þ 0:9333q�2 � 0:0757q�3 ;


GL2 ¼1

1� 0:2q�1 ¼ 1þ 0:21� 0:2q�1 q�1; F2 ¼ 1;

R2 ¼0:2

1� 0:2q�1 :

Since G1 has three unstable zeros, 2.8225, 1.1057 and �1.7839, wehave

Gþ1 ðq�1Þ ¼ ð1� 2:8225q�1Þð1� 1:1057q�1Þð1þ 1:7839q�1Þ;


¼ q�3ð1� 2:8225qÞð1� 1:1057qÞð1þ 1:7839qÞ

� 0:00971� 3:3516q�1 þ 4:4581q�2 � 2:7736q�3 þ 0:6671q�4

¼ 0:054

þ 0:1433� 0:2615q�1 þ 0:1595q�2 � 0:036q�3

1� 3:3516q�1 þ 4:4581q�2 � 2:7736q�3 þ 0:6671q�4 q�1;

eS ¼ 0:054; eT¼ 0:1433� 0:2615q�1 þ 0:1595q�2 � 0:036q�3

1� 3:3516q�1 þ 4:4581q�2 � 2:7736q�3 þ 0:6671q�4 ;

and

D ¼ q�nGþ1 ðqÞ=Gþ1 ðq�1Þ

¼ q�3ð1� 2:8225qÞð1� 1:1057qÞð1þ 1:7839qÞð1� 2:8225q�1Þð1� 1:1057q�1Þð1þ 1:7839q�1Þ ;

Thus,

DR1 ¼q�3ð1� 2:8225qÞð1� 1:1057qÞð1þ 1:7839qÞð1� 2:8225q�1Þð1� 1:1057q�1Þð1þ 1:7839q�1Þ

� 0:0991� 0:1294q�1 þ 0:0117q�2

1� 1:8575q�1 þ 0:9333q�2 � 0:0757q�3

¼ �0:0109� 0:1292q�1 þ 0:1215q�2

ð1� 2:8225q�1Þð1� 1:1057q�1Þð1þ 1:7839q�1Þ

þ 0:6809� 0:2394q�1 � 0:9551q�2

1� 1:8575q�1 þ 0:9333q�2 � 0:0757q�3 ;

Rnmp1 ¼ �0:0109� 0:1292q�1 þ 0:1215q�2

ð1� 2:8225q�1Þð1� 1:1057q�1Þð1þ 1:7839q�1Þ ;

Rmp1 ¼

0:6809� 0:2394q�1 � 0:9551q�2

1� 1:8575q�1 þ 0:9333q�2 � 0:0757q�3 :

0 0.5 1 1.5 2 2.5 30

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0.550.6

Variable k1

(a)

Perfo

rman

ce in

dex

(η)

Fig. 12. Control loop performance measure for


yt jmv ¼ F1a1t þRnmp

1

Da1tq�d1�d2 þ

eSD

a2tq�d1

¼ 0:1967q�1þ0:1546q�2þ �0:0109�0:1292q�1þ0:1215q�2

ð1�2:8225q�1Þð1�1:1057q�1Þð1þ1:7839q�1Þ

�ð1�2:8225q�1Þð1�1:1057q�1Þð1þ1:7839q�1Þð�2:8225þq�1Þð�1:1057þq�1Þð1:7839þq�1Þ q�3

�a1t

þ0:054 � ð1�2:8225q�1Þð1�1:1057q�1Þð1þ1:7839q�1Þð�2:8225þq�1Þð�1:1057þq�1Þð1:7839þq�1Þ q�2a2t ;


yt jmv ¼ ð0:1967q�1 þ 0:1546q�2 � 0:002q�3Þa1t þ ð0:0097q�2Þa2t :


2�1:3q�1

1�q�1 ; GC2 ¼ 0:5þ0:5q�1


is 0.3916. Consider GC1 ¼ 2�1:3q�1

1�q�1 ; GC2 ¼ k20:5þ0:5q�1



is 0.4202. Furthermore, consider GC1 ¼ k12�1:3q�1

1�q�1 ; GC2 ¼ k20:5þ0:5q�1

1�q�1 ,

the performance measure can be described as Fig. 13a and thecontour is also obtained as shown in Fig. 13b. The maximumperformance measure is about 0.5. Three cases also show thatthe control performance can be possibly improved.


G1 ¼0:0367ð1þ 0:0813q�1Þð1� 1:5378q�1 þ 0:9324q�2Þq�2

1� 3:2304q�1 þ 4:2278q�2 � 2:6431q�3 þ 0:6457q�4 ;

d1 ¼ 2;

G2 ¼q�1

1� 0:1q�1 ; d2 ¼ 1;

Gs ¼0:0618q�1 � 0:0766q�2

1� 1:7452q�1 þ 0:7452q�2 ; Gd ¼1

1� 0:1q�1 ;

GL1 ¼ GdGs

¼ ð0:0618q�1 þ 0:0374q�2Þ

þ 0:0122� 0:0298q�1 þ 0:0028q�2

1� 1:8452q�1 þ 0:9197q�2 � 0:0745q�3 q�3;

0 0.5 1 1.5 2 2.5 30

0.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0.550.6

Variable k2

Perfo

rman

ce in

dex

(η)

(b)

k1 and k2 at 50% power level, respectively.

0 0.5 1 1.5 2 2.5 3

00.511.5

22.53

00.10.20.30.40.50.6

Variable k1

(a)

Variable k2

Perfo

rman

ce in

dex

(η)

0.05

0.05

0.050.05

0.1

0.1

0.10.1

0.1

0.15

0.15

0.15

0.15

0.2

0.2

0.2

0.2

0.25

0.25

0.25

0.25

0.3

0.3

0.3

0.3

0.35

0.35

0.35

0.4

0.4

0.4

0.45

0.45

0.5

0.5

Variable k1

Varia

ble

k 2

(b)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3


0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

Variable k1

(a)

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

Variable k2

Perfo

rman

ce in

dex

(η)

(b)

Perfo

rman

ce in

dex

(η)


0 0.5 1 1.5 2 2.5

00.5

11.5

22.5

0

0.05

0.1

0.15

0.2

Variable k1

(a)

Variable k2

Perfo

rman

ce in

dex

(η)

0.02

0.02

0.020.02

0.04

0.04

0.040.04

0.06

0.06

0.060.06

0.08

0.08

0.08

0.08

0.1

0.1

0.1

0.1

0.12

0.12

0.12

0.14

0.14

0.14

Variable k1

Varia

ble

k 2

(b)

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5



F1 ¼ 0:0618q�1 þ 0:0374q�2; R1

¼ 0:0122� 0:0298q�1 þ 0:0028q�2

1� 1:8452q�1 þ 0:9197q�2 � 0:0745q�3 ;

GL2 ¼1

1� 0:2q�1 ¼ 1þ 0:21� 0:2q�1 q�1; F2 ¼ 1; R2 ¼

0:21� 0:2q�1 :

Since G1 has two unstable zeros, 1.2654 + 0.3294i and 1.2654–0.3294i, we have

Gþ1 ðq�1Þ ¼ 1� 1:5378q�1 þ 0:9324q�2;


¼ q�2ð1�1:5378q�1 þ 0:9324q�2Þ

� 0:0367ð1þ 0:0813q�1Þ1�3:2304q�1 þ 4:2278q�2 �2:6431q�3 þ 0:6457q�4

¼ 0:0342

þ 0:0568� 0:1125q�1 þ 0:0934q�2 � 0:0221q�3

1� 3:2304q�1 þ4:2278q�2 � 2:6431q�3 þ 0:6457q�4 q�1;


eS ¼ 0:0342; eT¼ 0:0568� 0:1125q�1 þ 0:0934q�2 � 0:0221q�3

1� 3:2304q�1 þ 4:2278q�2 � 2:6431q�3 þ 0:6457q�4 ;

and

D ¼ q�nGþ1 ðqÞ=Gþ1 ðq�1Þ ¼ q�2ð1� 1:5378qþ 0:9324q2Þð1� 1:5378q�1 þ 0:9324q�2Þ :

Thus,

DR1 ¼q�2ð1� 1:5378qþ 0:9324q2Þ1� 1:5378q�1 þ 0:9324q�2

� 0:0122� 0:0298q�1 þ 0:0028q�2

1� 1:8452q�1 þ 0:9197q�2 � 0:0745q�3

¼ 0:0035þ 0:0192q�1

1� 1:5378q�1 þ 0:9324q�2

þ �0:0374� 0:0078q�1 þ 0:006q�2

1� 1:8452q�1 þ 0:9197q�2 � 0:0745q�3 ;

Rnmp1 ¼ 0:0035þ 0:0192q�1

1� 1:5378q�1 þ 0:9324q�2 ;Rmp1

¼ �0:0374� 0:0078q�1 þ 0:006q�2

1� 1:8452q�1 þ 0:9197q�2 � 0:0745q�3 :


ytjmv ¼ F1a1t þRnmp

1

Da1tq�d1�d2 þ

eSD

a2tq�d1

¼ 0:0618q�1 þ 0:0374q�2 þ 0:0035þ 0:0192q�1

1� 1:5378q�1 þ 0:9324q�2

��1� 1:5378q�1 þ 0:9324q�2

0:9324� 1:5378q�1 þ q�2 q�3�

a1t

þ 0:0342 � 1� 1:5378q�1 þ 0:9324q�2

0:9324� 1:5378q�1 þ q�2 q�2a2t ;


ytjmv ¼ ð0:0618q�1 þ 0:0374q�2 þ 0:0038q�3Þa1t þ ð0:0367q�2Þa2t:


2:8�0:8q�1

1�q�1 ; GC2 ¼ 0:5þ0:5q�1


is 0.1057. Consider GC1 ¼ 2:8�0:8q�1

1�q�1 ; GC2 ¼ k20:5þ0:5q�1



is 0.1488. Consider GC1 ¼ k12:8�0:8q�1

1�q�1 ; GC2 ¼ k20:5þ0:5q�1


mance measure can be described as Fig. 15a and the contour is alsoobtained as shown in Fig. 15b. The maximum performance mea-sure is about 0.14. Three cases also show that the control perfor-mance can be possibly improved.

5. Conclusions and future work

A class of SG water level control systems, two PI controller sys-tems, is considered in the control loop performance assessment. Interms of the plant with stable and unstable zeros, control loop per-formance assessment methods are developed. The MV perfor-mance benchmark can be estimated by multivariate time seriesanalysis with a priori knowledge of process time delays for theplant with stable zeros and for the plant with unstable zeros con-sidering a priori knowledge of the locations of unstable zeros. Theydo not require perturbing the process with extraneous test signals.The numerical examples illustrates the effectiveness of the pro-posed method.

This paper presents a preliminary attempt of performanceassessment problem for the steam generator water level control

system based on the two PI controller structure. Some future worksinclude applying the proposed approach to a more realistic case inpractice about a controlled NPP, process identification and so on.There is a need to devote the future research efforts to a betterunderstanding of performance assessment for the steam generatorwater level control system through theoretical analysis as well asexperimental investigation.

Acknowledgements

The authors thank the Editors and the Reviewers for theirvaluable comments and suggestions. The authors gratefullyacknowledge the financial support, in part, of the Natural ScienceFoundation of China (Grant Nos. 60474020 and 60774006).

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