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Design and Application of Multivariable Control Systems in Power Plants
Copyright © (FAC Power Systems and Power Plant Control. Beijing, 1986
DESIGN AND APPLICATION OF MULTIVARIABLE CONTROL SYSTEMS IN POWER PLANTS Zhang Fa-wen Thermal Power Engineering R esearch Institute, Minist>)' of \\ '(I/er Eiee/I'ic Power, Xiall , PR C
RI'.IO llrU'.1
(Ill/I
Abstract. This paper describes a new method to evaluate diagonal dominance in a transfer matrix for a unilateral decoupled system. Three basic decourling metho ds are discussed. The control systems designed with these methods have been used in three units with once-through boilers and have good control qualities . Finally the ways to improve the availability of control systems are discussed.
Keywords. Multivariable control systems; power station control; decoupling; unilateral compensation; diagonal dominance.
Many methods can be used to design the multivari able control systems. Though the decoupling is frequently used but rather complicated. Full decoupling needs n(n-l) decoupling devices . Therefore. the method of unilateral decoupling has been worked out. which needs only n(n-l)/2 decoupling devices to reduce a multi variable system to single variable systems. Unilateral decoupling can improve the stability and quality of control systems. In Fig. 1 is given a comparision between control processes of the draft systems with and without a unilateral decoupling device for a 300 Nw generating unit.
without u. d. d.
0. %
reduced to many single variabl e systems . So the diagonal dominance of a de coupled syst em will be discussed first. DIAGONAL DOMINANCE OF UNILATERALLY DECOUPLED SYSTEl'S The block diagram of a multi variable control system is shown in Fig. 2.
with U.d.d Fig . 2.
Pro
The block diagram of a multivariable control system.
-5 m1ll
t 360 sec
tizO 0
Fig. 1.
ISO
According to decomposition theorem :
A comparison between control processes of the draft systems with and without a unilateral decoupling device (u.d.d.).
and defined as
The transfer matrix of a unilaterally decoupled equivalent system has the following form:
where h, is the transfer function seen between 1.
YI Y2
Yn where
Well we12 we22
r
w eln • we2n
l
w
enn
input i and output i when loops 1.2 ••••• i-1 are closed . with gains f .f ••••• f _ • and other loops l 2 i l are open. Then a following formul a can be deduced:
~ u 2
u
n When i=l
w . .the transfer function of the main el.l. channel. Generally. we can not implement the unilateral compensation in some of engineering practice. It can be done only approximately. Our research work shows. that under certain condition an approximate unilateral decoupled system can be diagonally dominant. thus a multi variable system is
So it is a single variabl e system. When i=2 det [IK + F
'w'J=
(l+flW ll ){ 1 + f 2 [w22 flW12w2l/(l+flwll)]}
355
356
Zhang Fa-\\·cn
The principal diagonal elements stand alo ne in above formula . The others are in form product w . w • One of these two elements is assumed t o 12 21 be w' , then we can calculate anothe r w. = 12 21 w12 . w21/ w·12 ' while the determinant value remains constant . For example , the following is a matrix at some frequency : 0. 1 [ 0 . 002
2l
e~J
lent channel . For series unil ateral decoupling the transfer funcof the compensator must be equal to: tion c ij
(1) where
a cofactor of determinant of matrix
w..
JJ
0. 4 ) 0. 2
J~
wit hout elements in rows fr om 1 to j - l a s well as in columns ; a cofactor of the element wji of
c. .
a transfer functi on, which usually
101 ••
determinant Wjj ;
I t is a non- diagonally dominant . Let w\2: w' 21 = 0 . 1 : 0 . 2 { w· • w. = 0 . 0008 12 21 The n the following values can be obtained: w\2 w'
w .. --- the transfer function of an equiva-
JJ
equals 1 .
2 . Feedback unilateral de coupling method The matrixes for this kind of decoupling are illustrated in Fi g . 4 .
0 . 02 0 .04
The matrix ( 0 .1 0 . 04
0 . 02 ] 0. 2
will be diagonally dominant, while its determinant remains constant . As mentioned above , there are n(n- l) non- principal diagonal elements , but only (n- l)(n- l) equations can be established, so (n-l) non- principal elements can be freely selected according to the diagonal dominance. Th:ough such selection of nonprincipal elements the matrix of unilateral decoupled system generally will become diagonal dominance. After such operation Gershgorin bands become narrower. It means that this method relaxes the requirement for diagonal dominance and makes implementing the decoupling system easier_
Fi g . 4 .
The block diagram of feedback unilateral decoupling .
Let
(I +
-,=r: .
1
21
Cb]
c '=
lc~ c
n2 • • • 1
1
THREE BASIC UNILATERAL DECOUFLING MEl'HODS
c'
Three basic unilateral de coupling methods have been summarized , as other unilateral decoupling methods can be ~ncluded in them . They are: 1. Series unilateral decoupling method The matrixes for this kind of decoupling are illustrated in Fig . 3.
C'
21
nl
1
c
1
n2
The elements of matrix C are calculated according to formul a (1) : 1
c "12
---El---GJFig . 3.
w ll w 2l
The block diagram of series unilateral de coupling. wl~
wln
101
w2n
cll c c 21 22
101
c
22
w wn2 nl
well we12 we22
nn
~J
c
n2
I
J
• we 1 n '), w e 2n
101
wher e c .. -
nl
1
C'=
enn
the transfer function of a compensater ;
Therefor
c. . ~J
e " ..
(2)
J~
i> j
3. Sum- and-difference method According to t he character of outputs from a multi variable system an operation of add or substraction is carried out to achieve the unil ateral compensation. The three methods are often used in mixed manner in accordance with features of t he transfer function.
In designing multivariable systems it is desired to put the zero element s on the left of the principal diagonal elements in t he matr ix to reduce the number of compensation devices to minimum.
Design and :\ppiicalioll where
357
M --- t he position of turbihe governor t valve;
AP?LICATro~
0 •.v-- fuel flow , feedwater flow and air flow ; D --- lead compensator.
IK roWER PLA..NTS
In 1972-1973 we applied the feedback unilateral decoupling method to design control systems for a high pressure once-through boiler ana obtained a good control quality (see Fig . 5) .
1 1
1
I
V
oil flow B I feedwater flow "F.D
Cl Fig .
5.
C<,
B
i
0" v.;, 101" W,W.
'T5+L 1
G,
G,
V
1
Gu
Sf
P
0
o
W;,'"
0
'i,
°2 S
f
0
~,O
o0
Fig .
0 «nO
7. The block diagram of cont r ol syst ems for a subcritical pressure once-throuth boiler .
9
Wll'¥t,.~W,l.JVl.ll
V"O WJJ~
°2
G.o
In 1974 we applie d the series and feedback unilateral decoupi ng methods to desi gn control sys tems for another once-through boiler and obtained a good quality ,t oo (see Fig . 6 ).
G1
h
Cl
The block diagram of open control systems for a high pressure oncethro ugh boiler.
LD l'i+pD
To implement the unilateral de coupling the fo llowing condition must be sati sfie d :
o Cl.! 0
o Fi g . 6 .
i ,e.
0 0
o 04,
0
block diagram of open control systems for ;"nnther once- through boiler.
~he
I n de signing the control systems a I - st lag device has been put in series with output from control device(as c ) to make the air flow change more ll q uickly than does the fuel flow . Between the control systems , for combustion and air flow control systems series and feedback unilateral compensations have been used , and feedback compensation for the other systems . To obtain a higher speed of res ponse of induction system , the compensation signal has oeen taken from the subloop of the airflow system for the purpose of improving the compensation e"ofects. In 1983 we used above three decoupling met hods to design t he multivariable control systems for a subcrit ical pressure once- through boiler \ see Fig. 7). The series and feedback decoupli ~~ signals are appli ed simultaneousl y from combustion system to the forced draft systems ,and feedback compensation signals fo r the other systems . The s~~-and-difference decoupling method is applied in re lat ionship between the turbine and boiler.
In adjusting the control system the parameters of the lead compensator were determined at first by theoretical evaluation and finally by adjustment in the power plant . The compensation curves obtained in test are gi ve n in Fig . 8 .
t
150 Fi g . 8
300 sec
Result of compensation with tne lead compensator.
Using t he sum-and- aifference decoupl i ng method , w12 .D will be added to t he transfer function w 22 of second system, thus the dynamic characte risti c of the main channel will be improved as shown in Fig. 9. I n order to change flue gas oxygen and steam enthalpy with load and to ensure the contrJ l quality , their set- point must be reset by the unit load demand and the signal from the first stage pressure of the turbine respectively.
358
Zhang Fa-wen
N
MW
~pD
310
80 Fig. 9.
IlD
Improving of the dynamic characteristic by sum-and-difference decoupling.
The load change of the 300 MW unit is given in Fig. 10. 300 MW
220 M'II p
a feedwater pressure h
o Fig. 10.
5
10
15
ZJ1
min
t
The load change curves of the 300 MW unit.
1500 T of standard fuel (7000 Kcal/Kg) can be saved every year as a result of steam conditions being kept at design values by the coordinated system.
designing control systems, as the first or the second system is used the single variable system which may not be switched on in operation, if it is possible. And the single variable system which suffers serious disturbance or works with a Dad control quality will be so arranged that it will not be used as n-th system, if it is possible. As for power plants, the load of turbogenerator is often changed, and the operation of adjusting synchronizer to match such load changes is a seri ous disturbance, which leads to change of steam pressure. So the steam press ure control system is designed as the first single variable system. For a control system designed with feedback unilateral decoupling method, whe n the steam pressure control system is changed over to manual operation, the compensation signals from this system are fed as feedforward signals to ot her single variable systems, which are able to match t his uisturbance with a good control quality. From above it is seen t hat the feedback unilateral decoupling has an advantage of feeding compensation signal continously from the system whi ch has been switched off. So we mainly use this method to uesign control systems. Ge nerally speaking, if i-th single variable control system ( usually the main controller of cascade system) is out of order, t he decoupl ing signal is fed to the input of subloop controller of the i-th system and the subloop can be switched on independently. This is aimed at getting higher availability of control systems. For example, on the 300 ~'II unit, the signal of enthalpy h from the feedwater s ystem sometimes can not be fed normally, thus the main controller can not work further. If the subloop controller can be switched on independently. In such case the coordinated control system of this 300 MW unit can still work , only resetting the set- point of the subloop controller sometimes is needed. In designing the control systems for power plants, some abnormal conditions of turbines and boilers must be taken into consiueration. For example, self os cillation of 300 M~ turbine governing system at load of 250 MW leads to oscillation of load and throttle pressure. In order to avoid t he oscilla_ tion of boiler control syst ems, the first Si ngle vari abl e system has been designed wit h sum-anddifference met hod, a nd it re ceives a load signal and a t hrot t l~ pressure differential signal. As a result, the combustion control systems can work nor mally in spite of t he oscillation of load and pressure. 'l'he tL'az.sient process of loa d change is shown in Fig. 11.
IMPROVI~ THE AVAILABILITY OF CONl'ROL SYSTElll ~
When decoupling is achieved, the multivariable sys tem has been reduced to many single variable systems. When one of them works abnormally or is out of order, the others will give poor performance or work with a low stability, even become unstable. So i t is very important in designing a multi variable control system. It is particularly important for power plants, where turbines and boilers operate with a poor controllability. The control systems designed with the unilateral de coupling method, meet such requirements better than those designed wit h other met hods. The control systems designed with unilateral decoupling method have better integrity. In case when from 1 to i-th single systems are switched off, i+l to n-th single variable systems work normally with original stability, because the characteristic ~quations for these systems do not contain any tranefer functions of 1 to i-th systems. So in
oil flow
300
Mw
t80 260
24
"'00
Fig. 11.
"0
,00
Curves showing no influence of turbine governor on the boiler master.
359
Design and Application
CONCLUSION
Unilateral compensation is a simple and practical method of decoupling a multivariable system. Unilateral decoupling is suitable for designing the control systems of power plants. 2. A transfer matrix of approximately unilateral decoupling may be equivalent to a matrix with diagonal dominance. with above method to calculate dominance Gershorin bands can be made narrower. So the requirment of diagonal dominance can be relaxed and the structure of control systems will be simpler. 3. Three basic unilateral decoupling methods are often used in mixed manner. The control systems designed with the feedback decoupling method are preferred in order to get high aVailability of control systems. The feedforward Signals are also needed to improve the control quality. 1.
REFERENCES
Mayne, D.Q. (1973). The design of linear multivariable systems. Automatica, Vol. lX. Mayne, D.Q. (1979). Sequential design of linear multi variable systems. Proc. lEE, Vol. 126, No.6. (1974). Computer-aid~d control Rosenbrock, H.H. Hodgson & Son Ltd., London. Chap. Bystem design. 3, pp. 138-145. Zhang Fa-wen (1981). Design and adjustment of unilateral noninteractive multi variable automatic klectric power, No.9. control systems.
DESIGN AND APPLICATION OF MULTIVARIABLE CONTROL SYSTEMS IN POWER PLANTS Zhang Fa-wen Thermal Power Engineering R esearch Institute, Minist>)' of \\ '(I/er Eiee/I'ic Power, Xiall , PR C
RI'.IO llrU'.1
(Ill/I
Abstract. This paper describes a new method to evaluate diagonal dominance in a transfer matrix for a unilateral decoupled system. Three basic decourling metho ds are discussed. The control systems designed with these methods have been used in three units with once-through boilers and have good control qualities . Finally the ways to improve the availability of control systems are discussed.
Keywords. Multivariable control systems; power station control; decoupling; unilateral compensation; diagonal dominance.
Many methods can be used to design the multivari able control systems. Though the decoupling is frequently used but rather complicated. Full decoupling needs n(n-l) decoupling devices . Therefore. the method of unilateral decoupling has been worked out. which needs only n(n-l)/2 decoupling devices to reduce a multi variable system to single variable systems. Unilateral decoupling can improve the stability and quality of control systems. In Fig. 1 is given a comparision between control processes of the draft systems with and without a unilateral decoupling device for a 300 Nw generating unit.
without u. d. d.
0. %
reduced to many single variabl e systems . So the diagonal dominance of a de coupled syst em will be discussed first. DIAGONAL DOMINANCE OF UNILATERALLY DECOUPLED SYSTEl'S The block diagram of a multi variable control system is shown in Fig. 2.
with U.d.d Fig . 2.
Pro
The block diagram of a multivariable control system.
-5 m1ll
t 360 sec
tizO 0
Fig. 1.
ISO
According to decomposition theorem :
A comparison between control processes of the draft systems with and without a unilateral decoupling device (u.d.d.).
and defined as
The transfer matrix of a unilaterally decoupled equivalent system has the following form:
where h, is the transfer function seen between 1.
YI Y2
Yn where
Well we12 we22
r
w eln • we2n
l
w
enn
input i and output i when loops 1.2 ••••• i-1 are closed . with gains f .f ••••• f _ • and other loops l 2 i l are open. Then a following formul a can be deduced:
~ u 2
u
n When i=l
w . .the transfer function of the main el.l. channel. Generally. we can not implement the unilateral compensation in some of engineering practice. It can be done only approximately. Our research work shows. that under certain condition an approximate unilateral decoupled system can be diagonally dominant. thus a multi variable system is
So it is a single variabl e system. When i=2 det [IK + F
'w'J=
(l+flW ll ){ 1 + f 2 [w22 flW12w2l/(l+flwll)]}
355
356
Zhang Fa-\\·cn
The principal diagonal elements stand alo ne in above formula . The others are in form product w . w • One of these two elements is assumed t o 12 21 be w' , then we can calculate anothe r w. = 12 21 w12 . w21/ w·12 ' while the determinant value remains constant . For example , the following is a matrix at some frequency : 0. 1 [ 0 . 002
2l
e~J
lent channel . For series unil ateral decoupling the transfer funcof the compensator must be equal to: tion c ij
(1) where
a cofactor of determinant of matrix
w..
JJ
0. 4 ) 0. 2
J~
wit hout elements in rows fr om 1 to j - l a s well as in columns ; a cofactor of the element wji of
c. .
a transfer functi on, which usually
101 ••
determinant Wjj ;
I t is a non- diagonally dominant . Let w\2: w' 21 = 0 . 1 : 0 . 2 { w· • w. = 0 . 0008 12 21 The n the following values can be obtained: w\2 w'
w .. --- the transfer function of an equiva-
JJ
equals 1 .
2 . Feedback unilateral de coupling method The matrixes for this kind of decoupling are illustrated in Fi g . 4 .
0 . 02 0 .04
The matrix ( 0 .1 0 . 04
0 . 02 ] 0. 2
will be diagonally dominant, while its determinant remains constant . As mentioned above , there are n(n- l) non- principal diagonal elements , but only (n- l)(n- l) equations can be established, so (n-l) non- principal elements can be freely selected according to the diagonal dominance. Th:ough such selection of nonprincipal elements the matrix of unilateral decoupled system generally will become diagonal dominance. After such operation Gershgorin bands become narrower. It means that this method relaxes the requirement for diagonal dominance and makes implementing the decoupling system easier_
Fi g . 4 .
The block diagram of feedback unilateral decoupling .
Let
(I +
-,=r: .
1
21
Cb]
c '=
lc~ c
n2 • • • 1
1
THREE BASIC UNILATERAL DECOUFLING MEl'HODS
c'
Three basic unilateral de coupling methods have been summarized , as other unilateral decoupling methods can be ~ncluded in them . They are: 1. Series unilateral decoupling method The matrixes for this kind of decoupling are illustrated in Fig . 3.
C'
21
nl
1
c
1
n2
The elements of matrix C are calculated according to formul a (1) : 1
c "12
---El---GJFig . 3.
w ll w 2l
The block diagram of series unilateral de coupling. wl~
wln
101
w2n
cll c c 21 22
101
c
22
w wn2 nl
well we12 we22
nn
~J
c
n2
I
J
• we 1 n '), w e 2n
101
wher e c .. -
nl
1
C'=
enn
the transfer function of a compensater ;
Therefor
c. . ~J
e " ..
(2)
J~
i> j
3. Sum- and-difference method According to t he character of outputs from a multi variable system an operation of add or substraction is carried out to achieve the unil ateral compensation. The three methods are often used in mixed manner in accordance with features of t he transfer function.
In designing multivariable systems it is desired to put the zero element s on the left of the principal diagonal elements in t he matr ix to reduce the number of compensation devices to minimum.
Design and :\ppiicalioll where
357
M --- t he position of turbihe governor t valve;
AP?LICATro~
0 •.v-- fuel flow , feedwater flow and air flow ; D --- lead compensator.
IK roWER PLA..NTS
In 1972-1973 we applied the feedback unilateral decoupling method to design control systems for a high pressure once-through boiler ana obtained a good control quality (see Fig . 5) .
1 1
1
I
V
oil flow B I feedwater flow "F.D
Cl Fig .
5.
C<,
B
i
0" v.;, 101" W,W.
'T5+L 1
G,
G,
V
1
Gu
Sf
P
0
o
W;,'"
0
'i,
°2 S
f
0
~,O
o0
Fig .
0 «nO
7. The block diagram of cont r ol syst ems for a subcritical pressure once-throuth boiler .
9
Wll'¥t,.~W,l.JVl.ll
V"O WJJ~
°2
G.o
In 1974 we applie d the series and feedback unilateral decoupi ng methods to desi gn control sys tems for another once-through boiler and obtained a good quality ,t oo (see Fig . 6 ).
G1
h
Cl
The block diagram of open control systems for a high pressure oncethro ugh boiler.
LD l'i+pD
To implement the unilateral de coupling the fo llowing condition must be sati sfie d :
o Cl.! 0
o Fi g . 6 .
i ,e.
0 0
o 04,
0
block diagram of open control systems for ;"nnther once- through boiler.
~he
I n de signing the control systems a I - st lag device has been put in series with output from control device(as c ) to make the air flow change more ll q uickly than does the fuel flow . Between the control systems , for combustion and air flow control systems series and feedback unilateral compensations have been used , and feedback compensation for the other systems . To obtain a higher speed of res ponse of induction system , the compensation signal has oeen taken from the subloop of the airflow system for the purpose of improving the compensation e"ofects. In 1983 we used above three decoupling met hods to design t he multivariable control systems for a subcrit ical pressure once- through boiler \ see Fig. 7). The series and feedback decoupli ~~ signals are appli ed simultaneousl y from combustion system to the forced draft systems ,and feedback compensation signals fo r the other systems . The s~~-and-difference decoupling method is applied in re lat ionship between the turbine and boiler.
In adjusting the control system the parameters of the lead compensator were determined at first by theoretical evaluation and finally by adjustment in the power plant . The compensation curves obtained in test are gi ve n in Fig . 8 .
t
150 Fi g . 8
300 sec
Result of compensation with tne lead compensator.
Using t he sum-and- aifference decoupl i ng method , w12 .D will be added to t he transfer function w 22 of second system, thus the dynamic characte risti c of the main channel will be improved as shown in Fig. 9. I n order to change flue gas oxygen and steam enthalpy with load and to ensure the contrJ l quality , their set- point must be reset by the unit load demand and the signal from the first stage pressure of the turbine respectively.
358
Zhang Fa-wen
N
MW
~pD
310
80 Fig. 9.
IlD
Improving of the dynamic characteristic by sum-and-difference decoupling.
The load change of the 300 MW unit is given in Fig. 10. 300 MW
220 M'II p
a feedwater pressure h
o Fig. 10.
5
10
15
ZJ1
min
t
The load change curves of the 300 MW unit.
1500 T of standard fuel (7000 Kcal/Kg) can be saved every year as a result of steam conditions being kept at design values by the coordinated system.
designing control systems, as the first or the second system is used the single variable system which may not be switched on in operation, if it is possible. And the single variable system which suffers serious disturbance or works with a Dad control quality will be so arranged that it will not be used as n-th system, if it is possible. As for power plants, the load of turbogenerator is often changed, and the operation of adjusting synchronizer to match such load changes is a seri ous disturbance, which leads to change of steam pressure. So the steam press ure control system is designed as the first single variable system. For a control system designed with feedback unilateral decoupling method, whe n the steam pressure control system is changed over to manual operation, the compensation signals from this system are fed as feedforward signals to ot her single variable systems, which are able to match t his uisturbance with a good control quality. From above it is seen t hat the feedback unilateral decoupling has an advantage of feeding compensation signal continously from the system whi ch has been switched off. So we mainly use this method to uesign control systems. Ge nerally speaking, if i-th single variable control system ( usually the main controller of cascade system) is out of order, t he decoupl ing signal is fed to the input of subloop controller of the i-th system and the subloop can be switched on independently. This is aimed at getting higher availability of control systems. For example, on the 300 ~'II unit, the signal of enthalpy h from the feedwater s ystem sometimes can not be fed normally, thus the main controller can not work further. If the subloop controller can be switched on independently. In such case the coordinated control system of this 300 MW unit can still work , only resetting the set- point of the subloop controller sometimes is needed. In designing the control systems for power plants, some abnormal conditions of turbines and boilers must be taken into consiueration. For example, self os cillation of 300 M~ turbine governing system at load of 250 MW leads to oscillation of load and throttle pressure. In order to avoid t he oscilla_ tion of boiler control syst ems, the first Si ngle vari abl e system has been designed wit h sum-anddifference met hod, a nd it re ceives a load signal and a t hrot t l~ pressure differential signal. As a result, the combustion control systems can work nor mally in spite of t he oscillation of load and pressure. 'l'he tL'az.sient process of loa d change is shown in Fig. 11.
IMPROVI~ THE AVAILABILITY OF CONl'ROL SYSTElll ~
When decoupling is achieved, the multivariable sys tem has been reduced to many single variable systems. When one of them works abnormally or is out of order, the others will give poor performance or work with a low stability, even become unstable. So i t is very important in designing a multi variable control system. It is particularly important for power plants, where turbines and boilers operate with a poor controllability. The control systems designed with the unilateral de coupling method, meet such requirements better than those designed wit h other met hods. The control systems designed with unilateral decoupling method have better integrity. In case when from 1 to i-th single systems are switched off, i+l to n-th single variable systems work normally with original stability, because the characteristic ~quations for these systems do not contain any tranefer functions of 1 to i-th systems. So in
oil flow
300
Mw
t80 260
24
"'00
Fig. 11.
"0
,00
Curves showing no influence of turbine governor on the boiler master.
359
Design and Application
CONCLUSION
Unilateral compensation is a simple and practical method of decoupling a multivariable system. Unilateral decoupling is suitable for designing the control systems of power plants. 2. A transfer matrix of approximately unilateral decoupling may be equivalent to a matrix with diagonal dominance. with above method to calculate dominance Gershorin bands can be made narrower. So the requirment of diagonal dominance can be relaxed and the structure of control systems will be simpler. 3. Three basic unilateral decoupling methods are often used in mixed manner. The control systems designed with the feedback decoupling method are preferred in order to get high aVailability of control systems. The feedforward Signals are also needed to improve the control quality. 1.
REFERENCES
Mayne, D.Q. (1973). The design of linear multivariable systems. Automatica, Vol. lX. Mayne, D.Q. (1979). Sequential design of linear multi variable systems. Proc. lEE, Vol. 126, No.6. (1974). Computer-aid~d control Rosenbrock, H.H. Hodgson & Son Ltd., London. Chap. Bystem design. 3, pp. 138-145. Zhang Fa-wen (1981). Design and adjustment of unilateral noninteractive multi variable automatic klectric power, No.9. control systems.