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AECL-8833
ATOMIC ENERGY ^ V ^ L'ENERGIEATOMIQUE
OF CANADA LIMITED V i D ^ F DU CANADA LIMITEE
THE APPLICATION OF MVPACK TO THE DESIGN OFMULTIVARIABLE CONTROL SYSTEMS FOR A
NUCLEAR STEAM GENERATOR
Emploi de MVPACK pour concevoir des systemesde commande multivariable pour les
chaudieres nucleaires
H.W. HINDS
Presented at the Power Plant Digital Control and Fault TolerantMicrocomputers Seminar, Phoenix, Arizona. 1985 Aprri 9 12
Chalk River Nuclear Laboratories Laboratoires nucleaires de Chalk River
Chalk River, Ontario
May 1985 mai
ATOMIC ENERGY OF CANADA LIMITED
THE APPLICATION OF MVPACK TO THE DESIGN OF MULTIVARIABLECONTROL SYSTEMS FOR A NUCLEAR STEAM GENERATOR
by
H.W. Hinds
Presented at the Power Plant Digital Control and Fault-TolerantMicrocomputers Seminar, Phoenix, Arizona, 1985 April 9-12
Reactor Control BranchChalk River Nuclear LaboratoriesChalk River, Ontario KOJ 1J0
1985 May
AECL-8833
L'ENERGIE ATOMIQUE DU CANADA, LIMITEE
Emploi de MVPACK pour concevoir des systèmes de commandemultivariable pour les chaudières nucléaires
par
H.W. Hinds
Résumé
Les systèmes de commande de la plupart des centrales nucléaires sont conçusau moyen de méthodes classiques monovariables qui traitent chaque boucle decommande indépendamment. Les interactions entre les boucles sont réaliséesempiriquement en réglant les contrôleurs de façon à obtenir les résultatsvoulus. Alors que la taille et la complexité des centrales augmentent etque ces dernières doivent fonctionner le plus près possible de leurcapacité maximale pour avoir un rendement économique, ces interactionsdynamiques prennent beaucoup d'importance et rendent peu fiablel'extrapolation des méthodes classiques. Cependant, un réglage améliorépeut être obtenu en ayant recours à des méthodes multivariables pour laconception et l'analyse des systèmes de commande. Ces méthodes considèrentla centrale fondamentalement comme un système d'interaction ayant denombreuses entrées et sorties rivalisant pour répondre à des objectifssouvent contradictoires.
MVPACK est un logiciel facilement utilisable ayant une forte capacité pourconcevoir et analyser des systèmes de commande multivariable, sous forme defonctions de transfert ou d'espace des états. Il est constitué par unensemble de modules interactifs que l'utilisateur voit comme unecalculatrice de niveau élevé. Il comprend une base de données, desméthodes d'interaction, une bibliothèque mathématique et une importantecollection d'algorithmes conceptuels comme ceux-ci: réduction de ladimension des systèmes matricials des modèles linéaires; positionnement depôles avec rétroaction des sorties; contrôle modal; contrôle optimal avecestimation aléatoires des états; et la méthode du critère de Nyquistinversé.
L'objet de ce rapport est la conception de contrôleurs multivariables pourles chaudières nucléaires. Les résultats disponibles du modèle du 15e
ordre sont la pression de la vapeur et le niveau d l'eau; les paramètres decommande sont le débit de la vapeur et le débit de l'eau d'alimentation.On décrit dans les grandes lignes quatre techniques conceptuelles parmicelles qui peuvent être employées dans le module MVPACK et on les appliqueau modèle.
Groupe de contrôle du réacteurLaboratoires nucléaires de Chalk River
Chalk River, Ontario KOJ 1J0
Mai 1985
AECL-8833
THE APPLICATION OF MVPACK TO THE DESIGN OF MULTIVARIABLECONTROL SYSTEMS FOR A NUCLEAR STEAM GENERATOR
H.W. Hinds*
ABSTRACT
Control systems for most nuclear power plants are designed using conventionalsingle-variable methods which treat each control loop independently. Interactionsbetween loops are accounted for empirically by tuning the controllers to achievethe desired results. As plants increase in size and complexity, and are requiredto operate closer to maximum capacity for greatest economic returns, these dynamicinteractions become more important and render the extrapolation of thesetechniques unreliable. However, improved regulation can be achieved when multi-variable design and analysis methods are used, which view the plant as a funda-mentally interacting system with numerous inputs and outputs vying to meet oftenconflicting objectives.
MVPACK is a user-friendly software package that combines a powerful capability todesign and analyze complex multivariable control systems, both in state space andtransfer function form, with an ease of application. MVPACK is a set of inter-active modules that appears to the user as a high-level calculator. It is com-posed of a database, interaction methods, a mathematical library, and an extensivecollection of design algorithms including: order reduction of linear models; poleshifting with output feedback; modal control; optimal control with stochasticstate estimation; and the inverse Nyquist array method.
The subject of this paper is the design of multivariable controllers for a nuclearsteam generator. The available outputs of the 15th-order model are steam pressureand water level; control inputs are steam and feedwater flows. Four of the designtechniques implemented in the MVPACK module are outlined and then applied to thismodel.
•Atomic Energy of Canada Ltd., Research Company, Chalk River Nuclear Laboratories,Chalk River, Ontario, Canada KOJ 1 JO
AECL-8833
NOMENCLATURE
AaBCDdFGH
IUJK
l,K2,Kk*LmNnPPon
qRr
stUuV
wx
y
GreekAK
AiX
9e
-a
plant dynamics matrixcoefficient of the open-loop characteristic polynomialplant input matrixplant output matrixdirect input-output matrixradii of Gershgorin circlesdiagonal feedback matrixplant open-loop transfer function matrix, dimension (mxp)(1) system closed-loop transfer function matrix,
dimension (mxm)(2) Kalman f i l t e r gainidentity matrixindicescost functioncontroller matrixfactors of Kvector (lxm) to convert outputs to a single outputpost-compensator matrixnumber of outputsnumber of encirclements of originnumber of statesnumber of inputsnumber of unstable open-loop poles(1) compensated open-loop frequency response matrix(2) output cost matrixvector (pxl) to convert single input into actual inputsinput cost matrix(1) reference value vector of dimension p or n(2) dominance ratiosLaplace variable (s"1)time (s)dual eigenvector matrix, ll^V"1input vector of dimension peigenvector matrix of Aweight vectorstate vector of dimension noutput vector of dimension m
working matrixVandermode matrix of the pre-assigned eigenvaluescharacteristic polynomial evaluated at \jeigenvalue (s"1)state derivative noise covariance matrixoutput noise covariance matrixdamping angleimaginary part of eigenvalue, angular velocity (rad.s"1)real part of eigenvalue (s"1)
INTRODUCTION
Control systems for most nuclear power plants have traditionally been designed byconventional single-variable methods which treat each control loop independently.Interactions between loops are accounted for empirically by tuning the controllersto achieve the desired results. The tuning may be done in the field by the opera-tors or by the designer using his intuition while working with a plant simula-tion. As plants increase in size and complexity and are required to operatecloser to the design values to maximize the economic returns, the dynamic inter-actions between loops become stronger and more limiting. The continued use of thetraditional techniques becomes unreliable and leads to economic penalties.
Multivariable design and analysis methods have been developed over the years buthave not achieved widespread practical use in the nuclear industry. These methodstreat the plant as a fundamentally interacting system with numerous inputs, out-puts and internal states. These design methods can be used to achieve specificresponse characteristics or to meet (at times conflicting) design objectives.
There are two reasons why the use of multivariable techniques is not morewidespread:
• lack of understanding by control engineers, and
• lack of suitable software support packages.
The software support is essential as it reduces the time and effort required tosolve an individual problem considerably, by freeing the designer from the detailsof mathematical algorithms and software generation. It also can reduce somewhatthe level of understanding necessary to apply a particular method, as many ofthese details can be built into the package. Thus it can help in building theunderstanding of the designers.
MVPACK is a software package developed at the Chalk River Nuclear Laboratoriesto aid in designing multivariable control schemes. It appears to the user as auser-friendly high-level calculator with a wide variety of design and analysisalgorithms as well as more basic mathematical operations.
The subject of this paper is the design, via four of the methods available InMVPACK, of multivariable controllers for a nuclear stem generator.
MULTIVARIABLE BASICS
The system under study is described by a l inear model about the operating point
chosen by the designer, with the state-space formulation
^ x = Ax + Bu (1)
y = Cx + Du (2)
A constant gain controller is usually expressed as
u = Kir-y) (3a)
for output feedback. Some control methods require state feedback instead, whichcan be expressed as
u = K(r-x) (3b)
Some of the design methods available use dynamic compensators instead of constant
gain compensators; however, these are not covered in detail here.
Eqs. 1 and 2 can be Laplace transformed and rearranged to give the input-outputtransfer function matrix for the plant
y(s) = G(s)u(s) (4)
where
G(s) = [C(sl-A)-1 B + D] (5)
The closed-loop response is given by
y(s) = H(s)r(s) (6)
where
H(s) = ( I + G(s)K)--1 G(s)K (7)
The block diagram of the complete plant with an output feedback con t ro l le r is
shown in Figure 1 .
The object of mu l t i var iab le design is to choose the values in K such that the
desired response character is t ics are achieved. These response charac ter is t ics may
be expressed in e i ther the time or frequency domains, for example:
• S t a b i l i t y . The closed-loop system should be asymptot ical lys tab le , that i s , a l l poles of the closed-loop system shouldl i e in the open l e f t ha l f -p lane.
• S t a b i l i t y margin. Let x be a pole of the closed-loop systemsuch that x = -a ± ju>; then the angle e = t a n " 1 a/m is ameasure of the s t a b i l i t y margin. Indeed, over a period2it/w. the damped o s c i l l a t i o n has a decrement equal toexp(-2110/10)- e should be larger than 45° (or a > 00) •
• Speed of response. The time taken for the system torespond to a step input indicates how e f fec t i ve l y i t w i l lfo l low changes of the inputs. Because speed of responsedepends on the posi t ion of the dominant poles, thesepoles should not be too close to the o r i g i n .
• S e n s i t i v i t y . The system's response should be insens i t i veto small perturbat ions or system parameter changes.
MVPACK
MVPACK is an in te rac t i ve computer-aided design package for mu l t i va r iab le control
systems. I t acts l i k e a high- level ca lcu la tor working with the fo l lowing types of
data:
CONTROLLER
PLANT
-*<g)— 1s
A
XC
Figure 1. Output-Feedback Controlleron a Multivariable Plant
• real matrices,
• polynomial matrices,
• rational polynomial matrices,
• frequency responses, and
• time histories.
It contains the following types of modules:
• utilities such as LOAD, EDIT, OUT, PLOT, BODE and NYQ,
• math library to perform standard matrix operations suchas ADD, SUBtract, MULtiply, INVerse, TRAnspose andEIGenvalues.
• high-level modules for
--order reduction--modal control—pole shifting—optimal control—Kalman filtering—state space to Laplace and frequency domain--inverse Nyquist array methods—simulation
The structure of the program is such that more modules may be added as required.Also, the program is designed to be user-friendly; extensive help messages areavailable at every stage in the design.
The user interacts with the design at various points by supplying data or choosingmethods. He can also iterate by repeating a design with modified parameters ifthe results of the previous iteration were not satisfactory.
NUCLEAR STEAM GENERATOR
The objective is to design a control system for the nuclear steam generator (NSG)described in Appendix A. It consists of a 15th order model (n=15) having twoinputs:
• steam throttle valve lift, and
• feedwater flow,
and two outputs:
• steam pressure, and
• downcomer level.
The open-loop frequency responses from inputs to outputs are shown in Figure 2.
Step responses are shown in Figure 3. Note that the i n i t i a l response to the step
input is often in the opposite direct ion to the longer term e f fec t . This non-
minimum phase character ist ic together with the fact that a l l four responses are
s ign i f i cant , leads to the problems commonly associated with steam generator
cont ro l .
The eigenvalues of this system are given in Table 1; the system is open-loop
unstable as the f i r s t eigenvalue is pos i t ive. Note that there is one uncontrol-
lable mode in the system associated with the primary water in le t plenum tempera-
ture and one unobservable mode associated with the primary out let plenun tempera-
ture. Elimination of these two variables leads to a 13th order system which is
f u l l y control lable and observable.
Number
123456789
101112131415
TABLE
STEAM GENERATOR
Eigenvalues
+2.47E-5-5.18E-2-0.239-0.331-0.886+j0.255 \-0 .886- j0 .255 )-1 .30- 1 . 4 3 *- 1 . 4 3 *-3 .20-3 .29-5.84-5.98+J0.0670 )-5.98-j0.0670 /-6.30
1
EIGENVALUES
Frequency(Hz)
0.147
0.952
D ampin
0.961
~1
*Asso"ciated with unobservable or uncontrol lable modes.
50
3,40
'Z 3015
20
- 90
! oFt
t
I -90
! -1800.001 0.01 0.1Frequency (Hz)
_ ™ - •
301 0.01
1
\\
s
0 . 1
—
1
30
20
10
0
-10
• L
• r ~" i
\\
0.001 0.01 0.1
90
0
-90
-180
-270
1 0.001 0.01 0.1Frequency (Hz)
30
20
10
0
-10
-200.001
\
\1 i
0.01 0.1
0.001 0.01 0.1Frequency (Hz)
NN
ss ^ -.
90
0
-90
-180
-270
———* __
- - • •
30
20
10
0
-10
-201 0.001 0.01 0.1
90
0
-90
-180
-270
1 0.001 0.01 0.1 1Frequency (Hz)
1
Figure 2. Open-Loop Frequency Response of the Nuclear Steam Generator
(a) G,, = Steam Pressure^ , . _ Steam Pressure , , „ Downcomer Level ... .v ' 11 Steam Valve Li f t {D> b1? " FppHwatPr FTT^T *C^ b ? i - ; t n J ; \t*u,a i ,-fr (d) E
Downcomer LevelFeedwater Flow
PRESSURE
DOWNCOMERLEVEL(a)
0
-100
-200
-300
0.2S
0.00
-0.2^
-0.60
20 40 80 60DOWNCOMERLEVEL
-20
c
1 .6
1 0
OS
0.0
; ^ - - ^
20 40 60 80 10
20 40 so ao
fiQurm 3o . Staam G i n t r a t o r Op«n-Loop R M p o m a
20 40 QO 80 100TIME la)
DESIGN METHODS
Modal Control
The principle of modal control is that the matrix K can be factored into 3 partsas
(8)
such that
• K2 converts the output y into modes
c Kd is a diagonal matrix of modal gains to shift the eigen-values of the modes = diagOc-f)
• Ki converts the resulting modal inputs into actual inputs.
Ideally, we would wish
I.UTBK 1
0
(9)
and
K2CV = [I ,0] (10)
These cannot in general be achieved exactly but suitable approximations areavailable (1), the simplest of which is
10
(111
(12)
where
are the first p columns of V, and
U consists of the first p rows of U and is suitable for m=p,p as in our case.
A more robust method is based on pseudo-inverses
K, = (BTB)"1 BTVm (13)
= UpCT(CCT)"1 (14)
but was not found to work as well in our NSG case.
With K\ and K2 defined by equations (11) and (12), trial and error was used toadjust Kd until the first two eigenvalues for the closed-loop were shifted asgiven in Table 2. The arrays Ki, Kj, K2 and K are given below:
Ki =-0.644
1.867
0.229
0.123K2 =
-4.61E-2
-0.432
0.732
-2.60E-2
* 0.02 I
K --1.38E-3
-2.78E-3
-9.55E-3
2.73E-2
The gain lq is as large as possible (within a factor of 2) without causing com-plex closed-loop poles to appear. The system is relatively insensitive to gain
11
l<2' Increases in ty cause the f i r s t eigenvalue (the one nominally contro l led
by Iq) to decrease while increasing only s l i g h t l y the second eigenvalue. The
value k2=ki was selected as a simple compromise.
Closed-loop simulation of th is system for i n i t i a l of fsets of the output variables
is shown in Figure 4 .
TABLE 2
CLOSED-LOOP POLES USING MODAL CONTROLLER
Number
123456789
10111213
Eigenvalue
-3.21E-2-5.64E-2-0.182-0.416-0.897+J0.256-0.897-j0.256-1.30-3.20-3.29-5.84-5 99+j6.96E--5.99-j6.96E--6.30
Frequency
0.148
0.953
Damping
0.962
STEAMVALVELIFT
STEAMPRESSUREfkPa)
— - : J
I0
\
\
-5o[0
0. 10L
0.06L
0 00 L /
0
2 0
2 0
2 0
4 0
40
~~ y — -
40TIME
6 0
SO
• 1 _
ao
8 0
so
- "CDS..
6 0
J.•
10
:
' 0
STEAMVALVELIFT(S)
FEEDHFLOW(t)
STEAMPRESSURE
12
Pole Shifting
Pole shifting is based on the principle that a cyclic system, which is completelycontrollable and observable, can be reduced to a single-input single-outputsystem. In practice, this means that the controller can be factored into 2 p :ts
K = qk* (15)
The vector q may be chosen either arbitrari ly or to approximately satisfy (2)
V"1 Bq = W (16)
where
W =M l
W 2
is a vector with no zero elements,
Wi = vector of dimension p corresponding to shifted poles,
W2 = vector of dimension n-p corresponding to non-shifted poles.
Normally, i f each element of W2 is chosen to be less than every element ofthen the n-p remaining poles wil l remain stable.
The vector k* is calculated as the solution of (2)
= A = (17)
where
C = companion form of C
_ ,nAi ~ xi " V in-1 n-2
" al
\- = preassigned eigenvalues
13
Applying this method to the steam generator with the two assigned poles at -0.016
and -0.070 gives the matrices
k* = [-4.681E-3 , 6.964E-4]
q =1
8.436
and the resulting controller
K =-4.68E-3 6.96E-4
-3.95E-2 5.88E-3
Closed-loop simulation of this system for in i t ia l offsets of the output variablesare shown in Figure 4.
Optimal Control
Optimal control is designed to minimize a quadratic cost function
GO
J = J [xT(t)Qx(t) + uT(t)Ru(t)]dt (18)
However, the controller produced requires that all states be fed back, and inpractice this is not possible. A Kalman filter can be designed to estimate allstates given the outputs only. The Kalman filter has the structure shown inFigure 5.
The details of the calculation of the controller matrix K and the Kalman filtergain matrix H are given in (3).
14
The Kalman filter design is based on a model of the plant. In this case, we chosea reduced, 6th order model which was computed using the MVPACK module RED (4_).The 6th order model was checked against the 15th order model and found to beidentical for practical purposes. We also chose noise covariance matricesrepresenting independent Gaussian noise of
H = diagUO3, 103, 106, 106, 106, 106)
0 = diag(l.l)
The Kalman filter gain is
H =
2.0953.8
850.2-13.6289.5
.797.2
180.62.09
49.0-996.4
8.57-0.739
PLANT
MEASUREMENT
CONTROL INPUTS
r~ 1
KALMAN
FILTERGAIN
if
PLANT MODEL
ESTIMATED STATES
Figure 5. Kalman Filter
15
The cost matrices selected are a reasonable compromise between the dynamics of the
states and the control actions. The chosen cost matrices are
Q = diag(6, 0.4, 0.04, 0.04, 0.04, 0.04)
R = diag(10", 10")
and the resulting state-feedback matrix is
K =-9.72x10-" -3.43xlO-3 -3.82xlO"3 0.211 -8.92xlO"3 -2.22xlO"3
2.12xlO-2 2.97x10-- -6.44x10-" -1.024 2.81xlO"2 -1.36X10"1*
When this controller is applied to the full model, the first 8 poles are-1.63xlO-2, -7.06xl0-2, -0.251, -0.363+J0.044, -0.902+J0.250, -1.30. Theclosed-loop response of this system is given in Figure 6. The response depends onwhether the Kalman filter has the correct initial conditions or not.
Inverse Nyquist Array Method
The inverse Nyquist array (INA) method is somewhat different from the othermethods described above as the controller matrix is chosen directly by thedesigner using an iterative approach rather than being the result of a calcula-tion. The inverse Nyquist array is a graphical tool that aids the designer in hischoice. Another major difference is that the INA method works in the frequencydomain with the input-output transfer matrix rather than in state space.
Suppose that we factor the controller into 3 portions, L, F, and Kj where F is adiagonal matrix.
By definition, Q(s) = (19)
Then the closed-loop transfer function is given by
H(s) = [I + - 1 Q(s) (20)
where we have, for convenience, placed some blocks in the feedback path as shownin Figure 7.
16
•TEAMVALVELIFT(X)
FEEDUATERFLOWIX)
STEAMPRESSURE(kPol
DOMNCOMERLEVEL
20 40
-60
(
0.04
0.02
0.00-0.02,
80 IOO
20 40 eo eo t o
40 00TIME (•)
Figur* So. Clo*sd-Loop R«»pon»»to I n i t i a l Offsat of 3t*am Pr*««ur*
STEAMVALVELIFT(X)
FEEOWATERFLOW(X)
STEAMPRESSURE(kPo)
DOWNCOMERLEVEL(m)
0.50
025
0.00
-0.25
-10
-15
-10
-20
..00°
0.76
0.E0
0.26
0.00
XOPTIMAL (KALMAN=O)
-OPTIMAL (KALMAN CORRECT)
F i gun 6b . C I os«d -Loop Rto I n i t i a l Off««t of Dounco
K,(s)u
G(s)y
L(s)
FIs)
Figure 7. General INA Design Model
Rewriting Eq. 20 in terms of inverses gives
H(s) = F + Q(s) (21)
where we use the notation
H(s) = H(s)"1 and Q(s) = Q(s)"1
17
The system s t a b i l i t y can be examined by p l o t t i n g the determinants | § ( s ) | andA
|H(s) | and counting the i r encirclements of the or ig in which we w i l l denote by Nj
and N2, respectively. The system is stable i f (5)
Mi - N2 = p0 (22)
where p0 = mmber of open-loop poles in the r ight half plane.
This c r i te r ion is a modified version of the standard Nyquist c r i t e r i on , but is not
useful in general. However, i f the system is diagonally dominant, then the number
of encirclements is equal to the sun of the encirclements by the individual diago-
nal elements.
Thus Ni and N2 can be determined for a diagonally dominant system by examingA A A A
the individual elements q-f-j and h-f-f. Note also that h-j-j = fi + qii andthus both Ni and N2 can be obtained from the same curve by shifting the criti-cal point to -fi from the origin.
Diagonal (row) dominance can be examined by plotting the Gershgorin circles whichhave radii
d7-(s) = 2 | q \ - j ( s ) l (23)
Provided these circles do not encircle the origin, the matrix Q(s) is diagonallydominant. These circles define a band about the q-f-i curves. The Gershgorincircles are dependent on the open-loop system only.
For our steam generator problem, over the frequency range 0.001 to 1 Hz, most INAplots had amplitudes less than 1. To obtain all curves >1, the following scalingwas performed (see Table 3):
ROW S, 1,100 scale row 1 by 100ROW S,2,10 scale row 2 by 10
Row scaling does not affect the diagonal (row) dominance. The resulting INA plotwith Gershgorin circles is shown in Figure 8a.
18
F i gure 8a. INA PIotafter Initial S c a l i n g
F i g u r e 8b. INA Plotafter A c h i e v i n g Dominance
TABLE 3
OPERATIONS TO ACHIEVE DIAGONAL DOMINANCE
MVPACK Command
ROW*P,i,j
R0W*S,i,a
ROW*A,i , j> a
ROW*C,i
R0W*NAME,I
QZE
Description
Permutation of rows i and j
Scaling row i with a constant, a
Adding a times row j to row i
Application of Rosenbrock's h i l lclimbfng algorftfm on row i
Premultiply 0 by matrix NAME( inver t NAME i f I specif ied)
Apply Q(0) to 0
•Similar commands for column operations are also available.
19
Note that the system is not diagonally dominant as, in general, q.^ > Pi»; theGershgorin circles of element (1,1) encircle the origin.
Diagonal dominance can be achieved by trial and error using the methods shown inTaule 3. Most of these are simple row and col win operations. The Rosenbrockhillclimb is an algorithm for minimizing the dominance ratio
(24)
where di(s) is given by Eq. 23, by performing a set of ROW A,i,j,a operations.
A A A
QZE is used to diagonaiize Q(s) as s-»O, by using Q(s) as s->0 for either K or L.It ensures diagonal dominance at low frequencies but, in general, has not beenfound useful.
To achieve diagonal dominance, we performed, by intuition, the following sequence
COL S,2,0.14
ROW C,l
ROW C,2
scale column 2 by 0.14
apply h i l lc l imbing to row 1
apply h i l lc l imbing to row 2
The result ing INA plot is shown in Figure 8b. Note that the system is now
diagonally dominant. The values of K and L are given by
K =0.01
0.00316
0.00293
0.101
L =
2Q
The final task is to choose the value of F = diag(f-j) which will give thedesired response. The problem is thus in the same form as that of modal control(Section 5.1) where
• <2 is equivalent to L
• Kj is equivalent to Ki
t Kd is equivalent to F
The gains of F were selected to be as high as possible without causing a damping
ratio to be less than 0.8. The gains found were
F = diag{-0.2, 0.06)
and thus the total controller gain is given by
-2.0E-3 -5.859E-4K =
1.356E-3 4.325E-2
The response to initial offsets in the output variables is shown in both Figures 4and 6.
Other Methods
Other, more elaborate, design methods are available in MVPACK but were notinvestigated as part of this report, namely,
o design and analysis of a dynamic compensator,
o design of an integral controller,
o pole shifting with constrained output feedback.
Steady-State Control
The controllers designed in the previous sections were all tested by considering arelaxation from an initial offset. In many cases, the response to changes in thesetpoint are of interest. The steady-state behaviour of the system is given by:
y = Hor (25)
21
where Ho = closed loop, steady-state gain matrix. Ideally Ho should be theidentity matrix, but in practice i t is considerably different; changes in thevariables do not approach the changes in the setpoints.
This could be corrected by
• integral action (a feature available in MVPACK which is notdiscussed in this report), and/or
by pre-multiplying by Ko-l
In the latter case, the system is represented as shown in Figure 9.
SUMMARY AND CONCLUSIONS
The paper describes the use of four basic controller design methods: modal, poleshifting, optimal, and inverse Nyquist array (INA), to design controllers for anuclear steam generator. The results from all four methods are presented in amanner which is directly comparable. A brief comparison of some peak values ispresented in Table 4. It appears from Table 4 that the INA method gives the bestcontroller, in that the peak crosstalk is less. Optimal control, however, is aclose second and could be improved further by a different choice of cost para-meters. However, this conclusion is very subjective and depends strongly on thecriteria of interest.
MVPACK was used as a tool in all the calculations and plotting shown in thispaper. It is very convenient to use and offers extensive "help" facilities to thenovice user. By having all the design methods on a common database and pro-gramming system, their intercomparison was simplified. The INA example illus-trates the convenience with which one can go from state-space to frequencyresponse and then back to time responses. It is a very versatile package.
STEADY-STATECORRECTOR
CONTROLLER
Figure 9. Correction for Steady-State Errors
22
TABLE 4
SUMMARY OF PERFORMANCE PARAMETERS
InitialOffset
SteamPressure
DowncomerLevel
DesignMethod
ModalPole Sh i f tOptimal (1)Optimal (2)INA
ModalPole Sh i f t i ngOptimal (1)Optimal (2)INA
PeakValve
21152
31000
SteamLift
%
.0
.0
.4
.0
.9
.1
.5
.4
.5
.3
Peak FeedwaterFlow
%
4.08.06.40.71.8
9.013.05.97.0
14.2
PeakCrosstalk
75 mm91 nrn58 mm49 mm43 mm
38.3 kPa22.2 kPa16.8 kPa10.9 kPa9.7 kPa
T D Kalman filter initialized to zero.
(2) Kalman filter initialized correctly.
REFERENCES
1 . P.D. McMorran and T.A. Cole, "Mul t ivar iable Control in Nuclear Power Stat ions:Modal Cont ro l " , Atomic Energy of Canada Limi ted, report AECL-6690,1979 December.
2. S. Mensah, P.D. McMorran, M. Pol is and W. Pask iev ic i , "Mul t ivar iableContro l ler for Nuclear Plants Based on Pole S h i f t i n g " , Atomic Energy of CanadaLimited, report AECL-7249, 1981 February.
3. M. Parent and P.D. McMorran, "Mul t ivar iable Control in Nuclear Power Stat ions:Optimal Cont ro l " , Atomic Energy of Canada L imi ted, report AECL-7244,1982 November.
4. M. Parent and P.D. McMorran, "Mul t ivar iable Control in Nuclear Power Stat ions:Order Reduction", Atomic Energy of Canada L imi ted, report AECL-7245,1982 December.
5. N. Roy, S. Mensah and J. Boisvert , "Design of a Mul t ivar iab le Contro l ler for aCANDU 600 MWe Nuclear Power Plant Using the INA Method", Atomic Energy ofCanada Limited, report AECL-8342, 1984 A p r i l .
23
APPENDIX A
STEAM GENERATOR MODEL (1)
State Variables
123456789
101112131415
Contro l
T i• Tpi• Tp2
Tp3
• Tpo
• Tm2• T|n3• Tm4
Ld• t-sl. ps
Xe• Td
Variables
Primary Water In le t Plenum TemperatureF i r s t Primary Water Lump TemperatureSecond Primary Water Lump TemperatureThird Primary Water Lump TemperatureFourth Primary Water Lump TemperaturePrimary Outlet Plenum TemperatureF i r s t Tube Metal Lump TemperatureSecond Tube Metal Lump TemperatureThird Tube Metal Lump TemperatureFourth Tube Metal Lump TemperatureDowncomer LevelSubcooled LengthSteam PressureBoi l ing Section ExitDowncomer Temperature
dCL Steam Valve Lift
2.WFi
Feedwater Flow
Output Variables
Ps and Ld-
STEAM GENERATOR MATRICES
MATRIX A
1
2
3
4
5
6
7
8
9
10
11
12
-1.4310
4.9450
0-5.6193E-02
00.6486
02.4243-02
00
0-9.8553E-02
0-1.5159E-02
0-5.397
02.2042E-02
02.6471E-02
04.7864E-02
00
-5.5930
0.76074.7474E-02
00
00.6281
00
2.4068.3261E-02
01.2807E-02
0-2.8646E-03
0-3.890
01.9680E-05
0-4.0438E-02
00
00
-1-7
00
03
00
0-0
2-1
04
02
0-4
05
.409
.0209E-02
.7607
.0290E-02
.1231
.406
.8940E-02
.2365E-03
.7539E-02
.9864E-02
.9803E-02
00
0-6
01
-10
4-2
00
01
00
2-6
0-0
0-1
0-0
.001
.970
.409
.5554
.945
.969
.890
.2906
.406
.5011E-02
.4226
.2821E-02
.9177
00
00
0-2
00
-51
10
06
00
00
20
0-5
02
•6219E-02
.593J.312E-02
.431
•O216E-O2
.4271
.4358
.406
.1165
.6503E-03
.2333E-02
00
00
0-0
00
06
-10
0-0
0-3
08
05
.1448
•2463E-02
.431
.2539
•9056E-02
.7362E-03
.6790E-02
0-15.88
00.1233
00
0.64860
4.7474E-020.2225
00
-2.0482E-02-9.5977E-02
00
-3.7881.123
1.2807E-026.0012E-02
-2.8646E-03-1.3424E-02
-1.8622E-020.6452
1.9680E-059.2223E-05
-4.0438E-02
-0.1895
0
0
0
0
2
0
-9
-5
3
2
2
4
.5924
.4243E-02
.8553E-02
.415
.3907E-03
.2O42E-02
.6471E-02
.7864E-02
STEAM GENERATOR MATRICES
MATRIX A (cont'd)
13
14
15
01.313
02.3680E-03
01.0688E-02
MATRIX B
123456789101112131415
002.4440-1.05504.2870.6594-0.1475-0.95b71.060-2.082
-63.28-4.8407E-020.4279
MATRIX C
1
2
00
00
09.7633E-04
0-1.1336E-O3
07.9461E-06
000.21510-9.2801E-0200.37725.8026E-02-1.2979E-02-8.4373E-020.6264-0.18322.317-5.4998E-03-1.106
00
00
0-5.6933E-04
06.9899E-04
03.0538E-03
00
01
0-2.427
0-2.6930E-02
0-2.1967E-02
00
00
0-0.4447
03.6157E-04
07.1755E-03
01
00
0-0.3819
0-0.3084
-5.837
00
00
9.4.
-1.-5.
7.-8.
00
00
7633E-045751E-03
1336E-033120E-03
9461E-062840E-02
1.313
2.3680E-03
1.0688E-02
0
0
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