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An Algorithm Applicable for Digital Speed Control of Water Turbines
Copvright © IL\C Digital Computer Applicatiolls to Process COlltrol. Yielllla, ;\ustria, I ~IH:;
AN ALGORITHM APPLICABLE FOR DIGITAL SPEED CONTROL OF WATER TURBINES H.-W. Muller Dr'jJa!lI//{' 1I1 0/ ,\/(,(//(/lIiml EII/{IIIf'f?'lIl/{, [ 'lIh l{'I:I/~)' 0/ B()r/IIII11, B(),/tlllll, r/?{;
Abstract. The use of micro-computers for digital speed control of water power plants facilitates the realization of complex control structures. Such a complex control structure is the compensation control. The basic principle of compensation controlling is based mainly on a realizable compensation of the system behaviour. The problem of controller design is in this way reduced to finding a suitable open loop control. To be introduced in this paper is a procedure named OWAN, which determines an optimal open loop control. In this connection, optimal control means, that the zeros of the transfer function are optimally chosen in the sense of a quadratic error integral. The optimization problem, whose calculation takes place only in the time domain, leads to a linear system of equations, which can be solved with the known methods. With the aid of OWAN, an "optimal" open loop control for the nonminimum-phase transfer function of a water power plant is determined and tested in simulation. Keywords. Computer control, Power system control.
I NTRODUCTI ON The development of hydroturbine governors was until recently restricted to partially empirically determined rules of adjustment for traditional structures as the temporary droop or double derivative governor. Only in the last years hydroturbine governors with complex structures were developed. The conventional analog realization of such governors, however, is relatively expensive. Therefore it is more economical to use micro-computers for the digital governing of hydroturbines. One type of controller, where the possible improvement of control behaviour often does not justify the expenditure of analog realization, is the compensation control.
Fig. 1
Control loop structure with a model of the controlled system
z._~
L:.:J
Design problem The old idea of the design problem is shown in Fig. 1. An undisturbed model of the controlled system is in parallel with the disturbed controlled system. If the value x is subtracted y
from the value x, only the value
Fig. 2
/I
la z
Equivalent block schematic
X
z will be in feedback. The behavior of the control loop could be represented by two open loops (Fig.2).
In realization of the algorithm the model of the controlled system becomes a component of the governor (Fig. 3) . 153
H ,- \\' , \lidlcr wit h an ~--------------,
w
I I
I I
The transfer function of the open loop control is
I I
R (s)
I I I I I I I ______________ JI
The design prob lem consists of the determination of suitable open loop controls Rw (s) and Rz (s). The transfer function is obtained by two simple relations (Fig. 2). (1)
I
S z (s) (1 - Rz (s) S (s)) .
(2 )
The ideal demands of a control design are (3 ) F (s) ~ and w (4 ) F (s) ~ 0 z With eq. (1) and (2 ) consequently the ideal open l oop control becomes Rw(S)
Rz(S)
= R(s)
zr
+ ... +d 1s 1
(s)
( 7)
is assumed as known. The choice of Nr(S) will be discussed later.
Realized control loop structure
RW ( s ). S (s)
=
wi th c k ~ 0 and 1 < k. For the present the polynomial Nr(S)
~
Fig. 3
0 and m .::. n.
I I I
I I
I I
~
= 1 /S (s)
(5 )
The first step of the procedure consists of the definition of a so-called "fictitious" system response Xo(s): Zs (s) Xo(s)
=
1
~. ~
s
(8)
Xe(s)
r
x
(s), with all parameters well known, s~ould fu lfill the following conditions: Xo (s) has a) no poles in the
right half plane and b) only single poles. The case, in which the second condition is not fulfilled, is discussed by MUller (1984). The "fictitious" system response x (t) results from Eq. (8) to 0 Av t An (9) + ••• Xo (t) = a 1 e with the eigen value A = 0 + w ( 10) ).J ).J ).J for x (t) .. impulse o(t), v = n+k e ( 11 ) and v = n+k+1 for xe(t) .. step o(t). (12)
But the open loop control R(s) must fulfill the following two realization conditions:
With Eqs. (7) and (8) th e system response XIs) is obtained to be
- R(s) can not have poles in the right half plane
x ( s ) = d X ( s ) +... + d lX (s) s 1 0 0 0
- the order of the denominator of R(s) must be smaller than the order of the nominator.
and in the time domain
Frank (1974) found one way to determine a technically realistic open loop control. In the following a procedure named OWAN is introduced, which determines an optimal open loop control. Optimal control means, in this connection, that the zeros of the transfer function are optimally chosen in the sense of a quadratic error integral. The optimization problem, whose calculation takes place only in the time domain, leads to a linear system of equations, which can be solved with the known methods.
x(t)=d 0 x (t)+ d1 (t) ... +dlx 0 (t) 00
THE PROCEDURE The transfer function S(s) is given to m b + b s + ... +bms Zs(s) 1 0 ( 6) S (s) = n a o + a 1 s + ... +a n s Ns (s)
*
(13)
(1)
(14)
1
(j) d, x (t) j=O J 0
=L
(15)
The coefficients do, ... ,d
should be l chosen to such a degree, that the function x(t) approximates the nominal function xsoll (t) optimally in the sense of the quadratic error integral I
=
J {x soll (t) - x(t) } 2 dt o F(do, ... ,d ) ~ min l
( 16)
If the function F(do, ... ,d ) has a l minimum and is differentiable, the optimal coefficients can be calculated from the 1+1 equations
Digital Speed COlltrol of \\'ater Turbilles
of b
0
~
=
(i
O, . . . ,l)
( 17)
1.
v
v
L
L
• 1\, i(d q
OF
o
od .
{j (x
od.
1.
1.
0
so
n(t)-x(t))2 dt } ! 0
~
{j
1.
0
1
(x
L
]
j =0
0
o
dt }
1\
+ •• + l l\ p
P
L
v
(t))
d , 1)
, 0
0
( 1 9)
v
L
L C'i
co
C'i {j e
p=l q=l q P v
0
L
;l(xsoll(t)-x(t))(- xo(t))
dt
j
(21)
(i)
dt- j xsoll(t)
o
0
The i-th derivation of Eq.(9)
xo(t) dt. (22)
r
=
s
=
is
i j
( 31)
p= 1
e p
p
(dO Aop + ... +d Apl) (25) l
is obtained. Eqs. x
o
and
(25)
( A + A )t q p dt }
0
o =
(32 )
A t C'i Ar - 1 e q }dt q q
v
{
L
q=l
~
r-l COj C'i A , x ll(t) q=l q q 0 so L
(33)
At e q
(34 )
dt
c
d
r,l+l 1
C'i
p=l
e
P
v
•L q= 1
The set of Eq. (17) matrix notation
A t p (d A0 1 + .. +dl A )' p o P A t e q
C'i Ai q q
dt
is
V
v
j
L
L
v
L
L
p=l q=l
q p
0
!
I
II
:,
a l + 1 , 1+1 !
dll
{j
:
I'
~l+lJ
(36 )
lA +A e
p
)t
q
dt }
0
i 0 1 · A (d A + .. +dl A ) q 0 p p
Inserting Eqs. (22) gives
d
I
~
q q
C'i u
i i
(27)
v
1 r
I.
(26)
. A t C'i A1.e q
o p=l q=l
thus obtained in
a 1 ,1+1
•
co
r
(35)
(t) dt =
L
0
e
+ .•• + a
v
r
J
{J
follows
0 co
C'i
Eq. (30) briefly (24)
( i)
jx(t)
Jx soll (t)
At C'i
L C'i
• Ar -1. As - 1 q p
(24)
and,with Eq.(15),after some transformation
L
v
p=l q=l q P
r,s
q=l
With
L
=
v
v
(30)
+ +
v
a
=
dt
q q
and the reductions
Xo (t)
x(t)
A t e q
C'i A
q=l
( i)
L
L
With the substitutions
(i)
xo(t)
xsoll (t)
o
o
ul
0
v
( i)
x(t)
( A + A )t . q p dt }A q1. Apl
C'i q C'i p {J e
p=l q=l
f
( A + A )t q p dt } Ai AOd + .. + q P 0
v
f
+
f
(29)
The i-th equation of the set of Eq. (17) is also a linear equation with the 1+1 unknown do, ... ,d : l
(t)-
2
(j)
d. x
soll
( A + A )t q p }dt·
At 'J C'i Ai e q 'dt f xsoll(t). q q o q=l
(18 )
and with Eq. (15)
6
e
q p 0
p=1 q=l
You get the i-th equation to
{f
C'i C'i
(28)
and
(24)
(28) into Eq.
The parameter optimization with the quadratic error integral leads consequently to a linear set of equations. Therefore,the integral (16) will be calculated without transformation in the frequency domain. The set of equations has the following characteristic attributes: - The (l+l)x (1+1) matrix is symmetrical to the principal diagonal. This result is derived - regarding the
H .-\\'. j\hillcr
J:i()
commutative law - directly from Eq. (32) by exchanging the subscripts r and s. This attribute saves storage place and computins time when programming.
(1 +3,855) (1 +6 r 95)
r
the eigen values ~ and the partial fraction coefficients a are in general complex. Mliller (1984) published the proof of this claim. Therefore the coefficients do, ... ,d are also l real, if the (1+1)y(1+1) matrix is regular.
(38)
1
(5 +0, 145 .: iO. 145) (5 +0.6)
- The matrix elements a tor elements c
and the vecr,s are real, although
1-7,75
5
is obtained. The nominal function xsoll (t) of Fiq.4 was proved to be favorable. x(l)
A program was developed, which mainly accomplishes the following tasks: - interactive reading-in of the "fictitious" system response X (5); o - computation of the coefficients a r,s and cr' There is no restriction to the nominal function x ll(t). so ' - computation and output of the coefficients do, ... ,d . l Subroutines of the program library RASP (Grlibel, 1983) were applied to the program.
Xu
1---.iI
Fig.4
Nominal function x so 1 1 (t)
The open loop control with the optimal denominator then results as R(s)
=
2 3 0,025+0,2325+0,2835 +0,775 2 3 0,025+0,2165+0,895 +5
(39)
Figure 5 shows the system response of the controlled system with and without the open loop control.
EXAMPLE With the aid of the procedure OWAN an optimal open loop control for a certain, strongly simplified model of a water power plant will be determined. With the parameteE of a concrete plant (Hoppe, 1981) the transfer function S (5)
tr--------------------------. .ct)
1-7,75 (1+3,855) (1+6,95) 1- TAS ( 37) (1+3,855) (1+6,95)
is obtained. This transfer function has one zero in the right half plain. This means, that the controlled system is an all-pass system. The poles of the open loop control R(s) are chosen with the following objectives in mind: in the first place the open loop control shall neutralize the influence of the positive zero on the system response. The duration of the influence corresponds to about twice the all-pass constant TA' Therefore the poles of the open loop control should chosen in a way, that - the appropriate to time functions fade out after the time t = 2T A
- in the half plane the poles lie to the left of the poles of the controlled system. Regarding these two aspects the "fictitious" system response
2S
Fig.5
30
35
\0
45
so
t/ .. c
~
System responses x(t), xist(t), x (t) soll
With the open loop control from Eg. (39) the control loop structure from figure 3 can be realized. The transfer function of the controlled system S(s) and of the model of the controlled system M(s) are equal: S(s)
M(s)
=
1-7,75 (1+3,855) (1+6,95)
Figures 6 and 7 show the system responses. The 1 indicates the results received with the subject design procedure compared with the responses marked with 2 as received with the conventional controller of the respective plant at optimal parameter setting.
Digilal Speed COI1lrol of Waler Turbines
15 7
Muller, H.-W. (1984). Algorithmen fur die digitale Drehzahlregelung von Wasserturbinen. Schriftenreihe des Lehrstuhls fur MeB- und Regelun9,..stechnj.js, Ruhr-Universitat Bochum, Heft 23. Frank, P. M. (1974). Entwurf von Regelkreisen mit vorgeschriebenem Verhalten. Karlsruhe, G.Braun Verlag. Grzybowski, R., Muller, H.-W. (1985). Ein Verfahren als Hilfsmittel zur Modellreduktion. Contributed to Automatisierungstechnik (at). To be published 1986. 10
Fig.6
15
20
2S
30
3S
100
105
SO
Step response w (t)
=
o tt)
IoS
so
Vue:
~
N
.;
la
Fig.7
15
20
2S
30
3S
100
Step response z (t)
t / . . c-'
o tt)
CONCLUSION A procedure named OWAN was presented. With this approach, as an example, an "optimal" open loop control for governing a water power plant was determined. The proposed procedure is, of course, applicable to any other controlled system and can as well be used for other problems. Thus, the application to model-order reduction was recently proposed (Grzybowski and Muller, 1985).
REFERENCES Grubel, G. (1983). Die regelungstechnische Programmbibliothek RASP. Regelungstechnik 1983, Heft 3, S. 75-81 . Hoppe, M. (1981). Die Regelung von Systemen mit AllpaB-Eigenschaften Dargestellt durch theoretische und experimentelle Untersuchung einer Wasserkraftanlage. Schriftenreihe des Lehrstuhls fur MeBund Regelungstechnik, Ruhr-Universitat Bochum, Heft 20.
DC A-F·
AN ALGORITHM APPLICABLE FOR DIGITAL SPEED CONTROL OF WATER TURBINES H.-W. Muller Dr'jJa!lI//{' 1I1 0/ ,\/(,(//(/lIiml EII/{IIIf'f?'lIl/{, [ 'lIh l{'I:I/~)' 0/ B()r/IIII11, B(),/tlllll, r/?{;
Abstract. The use of micro-computers for digital speed control of water power plants facilitates the realization of complex control structures. Such a complex control structure is the compensation control. The basic principle of compensation controlling is based mainly on a realizable compensation of the system behaviour. The problem of controller design is in this way reduced to finding a suitable open loop control. To be introduced in this paper is a procedure named OWAN, which determines an optimal open loop control. In this connection, optimal control means, that the zeros of the transfer function are optimally chosen in the sense of a quadratic error integral. The optimization problem, whose calculation takes place only in the time domain, leads to a linear system of equations, which can be solved with the known methods. With the aid of OWAN, an "optimal" open loop control for the nonminimum-phase transfer function of a water power plant is determined and tested in simulation. Keywords. Computer control, Power system control.
I NTRODUCTI ON The development of hydroturbine governors was until recently restricted to partially empirically determined rules of adjustment for traditional structures as the temporary droop or double derivative governor. Only in the last years hydroturbine governors with complex structures were developed. The conventional analog realization of such governors, however, is relatively expensive. Therefore it is more economical to use micro-computers for the digital governing of hydroturbines. One type of controller, where the possible improvement of control behaviour often does not justify the expenditure of analog realization, is the compensation control.
Fig. 1
Control loop structure with a model of the controlled system
z._~
L:.:J
Design problem The old idea of the design problem is shown in Fig. 1. An undisturbed model of the controlled system is in parallel with the disturbed controlled system. If the value x is subtracted y
from the value x, only the value
Fig. 2
/I
la z
Equivalent block schematic
X
z will be in feedback. The behavior of the control loop could be represented by two open loops (Fig.2).
In realization of the algorithm the model of the controlled system becomes a component of the governor (Fig. 3) . 153
H ,- \\' , \lidlcr wit h an ~--------------,
w
I I
I I
The transfer function of the open loop control is
I I
R (s)
I I I I I I I ______________ JI
The design prob lem consists of the determination of suitable open loop controls Rw (s) and Rz (s). The transfer function is obtained by two simple relations (Fig. 2). (1)
I
S z (s) (1 - Rz (s) S (s)) .
(2 )
The ideal demands of a control design are (3 ) F (s) ~ and w (4 ) F (s) ~ 0 z With eq. (1) and (2 ) consequently the ideal open l oop control becomes Rw(S)
Rz(S)
= R(s)
zr
+ ... +d 1s 1
(s)
( 7)
is assumed as known. The choice of Nr(S) will be discussed later.
Realized control loop structure
RW ( s ). S (s)
=
wi th c k ~ 0 and 1 < k. For the present the polynomial Nr(S)
~
Fig. 3
0 and m .::. n.
I I I
I I
I I
~
= 1 /S (s)
(5 )
The first step of the procedure consists of the definition of a so-called "fictitious" system response Xo(s): Zs (s) Xo(s)
=
1
~. ~
s
(8)
Xe(s)
r
x
(s), with all parameters well known, s~ould fu lfill the following conditions: Xo (s) has a) no poles in the
right half plane and b) only single poles. The case, in which the second condition is not fulfilled, is discussed by MUller (1984). The "fictitious" system response x (t) results from Eq. (8) to 0 Av t An (9) + ••• Xo (t) = a 1 e with the eigen value A = 0 + w ( 10) ).J ).J ).J for x (t) .. impulse o(t), v = n+k e ( 11 ) and v = n+k+1 for xe(t) .. step o(t). (12)
But the open loop control R(s) must fulfill the following two realization conditions:
With Eqs. (7) and (8) th e system response XIs) is obtained to be
- R(s) can not have poles in the right half plane
x ( s ) = d X ( s ) +... + d lX (s) s 1 0 0 0
- the order of the denominator of R(s) must be smaller than the order of the nominator.
and in the time domain
Frank (1974) found one way to determine a technically realistic open loop control. In the following a procedure named OWAN is introduced, which determines an optimal open loop control. Optimal control means, in this connection, that the zeros of the transfer function are optimally chosen in the sense of a quadratic error integral. The optimization problem, whose calculation takes place only in the time domain, leads to a linear system of equations, which can be solved with the known methods.
x(t)=d 0 x (t)+ d1 (t) ... +dlx 0 (t) 00
THE PROCEDURE The transfer function S(s) is given to m b + b s + ... +bms Zs(s) 1 0 ( 6) S (s) = n a o + a 1 s + ... +a n s Ns (s)
*
(13)
(1)
(14)
1
(j) d, x (t) j=O J 0
=L
(15)
The coefficients do, ... ,d
should be l chosen to such a degree, that the function x(t) approximates the nominal function xsoll (t) optimally in the sense of the quadratic error integral I
=
J {x soll (t) - x(t) } 2 dt o F(do, ... ,d ) ~ min l
( 16)
If the function F(do, ... ,d ) has a l minimum and is differentiable, the optimal coefficients can be calculated from the 1+1 equations
Digital Speed COlltrol of \\'ater Turbilles
of b
0
~
=
(i
O, . . . ,l)
( 17)
1.
v
v
L
L
• 1\, i(d q
OF
o
od .
{j (x
od.
1.
1.
0
so
n(t)-x(t))2 dt } ! 0
~
{j
1.
0
1
(x
L
]
j =0
0
o
dt }
1\
+ •• + l l\ p
P
L
v
(t))
d , 1)
, 0
0
( 1 9)
v
L
L C'i
co
C'i {j e
p=l q=l q P v
0
L
;l(xsoll(t)-x(t))(- xo(t))
dt
j
(21)
(i)
dt- j xsoll(t)
o
0
The i-th derivation of Eq.(9)
xo(t) dt. (22)
r
=
s
=
is
i j
( 31)
p= 1
e p
p
(dO Aop + ... +d Apl) (25) l
is obtained. Eqs. x
o
and
(25)
( A + A )t q p dt }
0
o =
(32 )
A t C'i Ar - 1 e q }dt q q
v
{
L
q=l
~
r-l COj C'i A , x ll(t) q=l q q 0 so L
(33)
At e q
(34 )
dt
c
d
r,l+l 1
C'i
p=l
e
P
v
•L q= 1
The set of Eq. (17) matrix notation
A t p (d A0 1 + .. +dl A )' p o P A t e q
C'i Ai q q
dt
is
V
v
j
L
L
v
L
L
p=l q=l
q p
0
!
I
II
:,
a l + 1 , 1+1 !
dll
{j
:
I'
~l+lJ
(36 )
lA +A e
p
)t
q
dt }
0
i 0 1 · A (d A + .. +dl A ) q 0 p p
Inserting Eqs. (22) gives
d
I
~
q q
C'i u
i i
(27)
v
1 r
I.
(26)
. A t C'i A1.e q
o p=l q=l
thus obtained in
a 1 ,1+1
•
co
r
(35)
(t) dt =
L
0
e
+ .•• + a
v
r
J
{J
follows
0 co
C'i
Eq. (30) briefly (24)
( i)
jx(t)
Jx soll (t)
At C'i
L C'i
• Ar -1. As - 1 q p
(24)
and,with Eq.(15),after some transformation
L
v
p=l q=l q P
r,s
q=l
With
L
=
v
v
(30)
+ +
v
a
=
dt
q q
and the reductions
Xo (t)
x(t)
A t e q
C'i A
q=l
( i)
L
L
With the substitutions
(i)
xo(t)
xsoll (t)
o
o
ul
0
v
( i)
x(t)
( A + A )t . q p dt }A q1. Apl
C'i q C'i p {J e
p=l q=l
f
( A + A )t q p dt } Ai AOd + .. + q P 0
v
f
+
f
(29)
The i-th equation of the set of Eq. (17) is also a linear equation with the 1+1 unknown do, ... ,d : l
(t)-
2
(j)
d. x
soll
( A + A )t q p }dt·
At 'J C'i Ai e q 'dt f xsoll(t). q q o q=l
(18 )
and with Eq. (15)
6
e
q p 0
p=1 q=l
You get the i-th equation to
{f
C'i C'i
(28)
and
(24)
(28) into Eq.
The parameter optimization with the quadratic error integral leads consequently to a linear set of equations. Therefore,the integral (16) will be calculated without transformation in the frequency domain. The set of equations has the following characteristic attributes: - The (l+l)x (1+1) matrix is symmetrical to the principal diagonal. This result is derived - regarding the
H .-\\'. j\hillcr
J:i()
commutative law - directly from Eq. (32) by exchanging the subscripts r and s. This attribute saves storage place and computins time when programming.
(1 +3,855) (1 +6 r 95)
r
the eigen values ~ and the partial fraction coefficients a are in general complex. Mliller (1984) published the proof of this claim. Therefore the coefficients do, ... ,d are also l real, if the (1+1)y(1+1) matrix is regular.
(38)
1
(5 +0, 145 .: iO. 145) (5 +0.6)
- The matrix elements a tor elements c
and the vecr,s are real, although
1-7,75
5
is obtained. The nominal function xsoll (t) of Fiq.4 was proved to be favorable. x(l)
A program was developed, which mainly accomplishes the following tasks: - interactive reading-in of the "fictitious" system response X (5); o - computation of the coefficients a r,s and cr' There is no restriction to the nominal function x ll(t). so ' - computation and output of the coefficients do, ... ,d . l Subroutines of the program library RASP (Grlibel, 1983) were applied to the program.
Xu
1---.iI
Fig.4
Nominal function x so 1 1 (t)
The open loop control with the optimal denominator then results as R(s)
=
2 3 0,025+0,2325+0,2835 +0,775 2 3 0,025+0,2165+0,895 +5
(39)
Figure 5 shows the system response of the controlled system with and without the open loop control.
EXAMPLE With the aid of the procedure OWAN an optimal open loop control for a certain, strongly simplified model of a water power plant will be determined. With the parameteE of a concrete plant (Hoppe, 1981) the transfer function S (5)
tr--------------------------. .ct)
1-7,75 (1+3,855) (1+6,95) 1- TAS ( 37) (1+3,855) (1+6,95)
is obtained. This transfer function has one zero in the right half plain. This means, that the controlled system is an all-pass system. The poles of the open loop control R(s) are chosen with the following objectives in mind: in the first place the open loop control shall neutralize the influence of the positive zero on the system response. The duration of the influence corresponds to about twice the all-pass constant TA' Therefore the poles of the open loop control should chosen in a way, that - the appropriate to time functions fade out after the time t = 2T A
- in the half plane the poles lie to the left of the poles of the controlled system. Regarding these two aspects the "fictitious" system response
2S
Fig.5
30
35
\0
45
so
t/ .. c
~
System responses x(t), xist(t), x (t) soll
With the open loop control from Eg. (39) the control loop structure from figure 3 can be realized. The transfer function of the controlled system S(s) and of the model of the controlled system M(s) are equal: S(s)
M(s)
=
1-7,75 (1+3,855) (1+6,95)
Figures 6 and 7 show the system responses. The 1 indicates the results received with the subject design procedure compared with the responses marked with 2 as received with the conventional controller of the respective plant at optimal parameter setting.
Digilal Speed COI1lrol of Waler Turbines
15 7
Muller, H.-W. (1984). Algorithmen fur die digitale Drehzahlregelung von Wasserturbinen. Schriftenreihe des Lehrstuhls fur MeB- und Regelun9,..stechnj.js, Ruhr-Universitat Bochum, Heft 23. Frank, P. M. (1974). Entwurf von Regelkreisen mit vorgeschriebenem Verhalten. Karlsruhe, G.Braun Verlag. Grzybowski, R., Muller, H.-W. (1985). Ein Verfahren als Hilfsmittel zur Modellreduktion. Contributed to Automatisierungstechnik (at). To be published 1986. 10
Fig.6
15
20
2S
30
3S
100
105
SO
Step response w (t)
=
o tt)
IoS
so
Vue:
~
N
.;
la
Fig.7
15
20
2S
30
3S
100
Step response z (t)
t / . . c-'
o tt)
CONCLUSION A procedure named OWAN was presented. With this approach, as an example, an "optimal" open loop control for governing a water power plant was determined. The proposed procedure is, of course, applicable to any other controlled system and can as well be used for other problems. Thus, the application to model-order reduction was recently proposed (Grzybowski and Muller, 1985).
REFERENCES Grubel, G. (1983). Die regelungstechnische Programmbibliothek RASP. Regelungstechnik 1983, Heft 3, S. 75-81 . Hoppe, M. (1981). Die Regelung von Systemen mit AllpaB-Eigenschaften Dargestellt durch theoretische und experimentelle Untersuchung einer Wasserkraftanlage. Schriftenreihe des Lehrstuhls fur MeBund Regelungstechnik, Ruhr-Universitat Bochum, Heft 20.
DC A-F·