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A Realization Method of the Transfer Functions Containing Variable Parameter
Copyright © IFAC System Structure and Control, Nantes, France, 1995
A REALIZATION METHOD OF THE TRANSFER FUNCTIONS CONTAINING VARIABLE PARAMETER ATSUSHI KAWAKAMI Kanazawa Institute of Technology, Department of Electronics, 7-1, Ougigaoka, Nonoichi-machi, Ishikawa-ken, 921 Japan Abstract. In this paper, we propose a method for realizing transfer functions containing variable parameter, by the state-space method. By using this method, variable transfer functions (VTF) can be often realized with a minimal dimension. In case that a minimal realization can not be obtained, the realization dimension can be fairly reduced. Key Words. Variable parameter; variable transfer functions; statical feedback; minimal realization; dynamical feedback
Some methods for realizing VTF have been reported (Newcomb, 1968; Delansky, 1969) and they are very interesting. But, these methods have the difficulty that the realization dimension is very high. Hence, the circuit synthesis is very complicated.
1. INTRODUCTION Circuits containing variable elements are the general name of the circuits containing elements whose values can be varied independently to time and frequency, by varing the values continuously, the designated input-output characteristics can be obtained continuously. Such variable circuits take an important place in the measurement-control system and the communication system. Especially, in the design of the variable equalizers, many researches have been done so far (Bode, 1938 et ~)
Therefore, in this paper, in order to overcome the difficulty of the above-mentioned methods, we propose a method for realizing VTF by the state-space method. In this method, we perform the statical feedback on the state-space realization form of separabledenominator VTF, so that we realize the general VTF. By using this method, VTF can be often realized with a minimal dimension.
.
Though, various methods to synthesize the circuits containing variable elements mentioned above exist, the synthesis method in terms of variable transfer functions (VTF) which consist of the complex angular frequency s and the variable parameter A is thought to be an interesting method from the point of view of the unification of the theory.
In case that minimal realization can not be obtained, we propose a method that we perform the dynamical feedback on it. By using this method, the realization dimension can be fairly reduced.
67
2. VOLTAGE TRANSFER FUNCTIONS CONTAINING VARIABLE PARAMETER Circuit functions of circuit containing m gearing variable elements, can be expressed by
4. MINIMAL REALIZATION OF VTF In (1), it is assumed that n~m. Similarly, we can realize it in case that n
(Kawakami et aI, 1982a). We call this F(s,A) variable transfer functions (VTF).
4.1. Modification of VTF We perform a bilinear transformation of A in the following
3. STATE-SPACE REALIZATION FORM OF SEPARABLE-DENOMINATOR VTF
Separable-denominator VTF in the following, can be realized with a minimal dimension (Kawakami et aI, 1982b). FS (s ,A) =g (s ,A) / {a (s) . b CA.}}
A = (cA +d) / (aA +b)
{9}
on F(s,)") 1n (1). And, we decide values of a, b, c, d so that ~m(s) of
(3)
A minimal dimension is a degree with respect to s, A, of its denominator. is (n-1)-degree polynomial and 'f'm(s) of (10) is n-degree polynomial.
FS (s , A) in (3 ) can be rea 1 i zed by the following two types of statespace form, by the method in (Kawakami ~, 1982b).
Dividing the denominator and numerator of £(s,A) in (10) by the coefficient of srt of tm(s), we obtain
Controllable-Observable (C-O) state-space realization form:
A
mN
I
\IL
\/~
J!l.N
F(S,A)=Lg·(s)1\. / Lfi(s)/\. i=o (, i=o (11 ) =
where
Ln. g. (A: ) s J. / Lrt_f j. (X) s J. . I
J.=o
d-
I
J.=o
(12 )
Y/i = (Y,--- Yml T, U1L = (u,--- umlT
(5)
Yi=>",ui
(6)
4.2. Realization of SeparableDenominator VTF Containing Unknown Quantities
(7 )
Consider the case of
Ib 2. = (
f)
(i=I,2,-----,m) 1IT.
Observable-Controllable (O-C) state-space realization form:
68
where . a~ (i=O,I,---,n-I) are unknown quant1t1es.
From (17) and (18), equation (19) can be expressed as /V
We obtain the C-O state-space • \' realizat10n form of AFS { s, ....... ,all} 1n (13). It can be expressed as
-
/
F (s , X , al·) =
[,
(20)
~/V \ li /V g • (s).I\. I { f m{s } L=o ~ L
_
1
\1
A
·fn(A:)+h(s,1\. ,al,}} .
+ CIb~ i Ib2 )T U. T
I; rL2)()[Tj/U{)T.
\/
Then, VTF of the system in (16) is
t~J =t~~~j!1 ~;~6lj!] ~~Z] ~=(e)
A
/
fm(s)fn(X)+h(s,.I\.,ali)'
(21)
We express denominator polynomial of Y(s,>(} in (II) as
0+)
,IV
_
1
/
fm(s)fn(X}+h(s,X}. 4.3. Statical Feedback Containing Unknown Quantities
(22)
Comparing (22) with (20), we calculate the values of ai (i=O, I, --, n-I) so that
We perform the following statical feedback on (14).
(23) u =If
(15)
(al't, ) ]1+ V
In case that such aj. (i=O, 1,---,n-1) don't exist, it is impossible to realize F(s,}.J in (I) with a minimal dimension, by this method. Then, we apply the realization method which we will propose 1n 5.
If (al· ) in (IS) can be ca 1 c u 1 ate d by the "method in (Bass ~, 1965). Then, the following system can be obtained.
Now, it is assumed that such a' (i=O,I,--,n-l) (equation (23) i~ satisfied) exist. Substituting those aL into (16), equation (16) is a minimal realization of Y(s,X) in (II).
<
In case that n m in F (s,>..) in (I), we consider a separabledenominator VTF in the following.
Now,
I ~/V \IL "" Fs (s ,A , al./, ) = .L g L. (s) I\. I { fm (s )
,IV
A
} = f m(s)
de t
{s En-lA II ( all.)
det
(AEm-lA 22)=fn (A)
A
(17)
£,=0 m-I
L=o
We calculate the following characteristic polynomial of (16),
A
5. I. Modification of VTF
(all.) -Ib IIf (aiL)} • de t (A.'Ern
\/
t,
5. REALIZATION METHOD BY DYNAMICAL FEEDBACK
det[-~~t1~ ~!I_(~i)j __ -~ AI?:~F}:~)J - fA21 (~L) l X'][m- IAZl -/A Z2 )+h(s,.I\. ,aiL)'
+AIm )} (24)
We obtain O-C state-space realization form of Ys(s,,(,aIL) in (24), and after that, we calculate the values of al (i=O, 1,---, m-I) through the simIlar steps.
4.4. Calculation of Unknown Quantities
JI
It,
·(La.A.
are satisfied.
=det {s~-/A
•
~
( 18)
We perform a bilinear
(19) 69
transformation of A in the following 11
A=(gA
'P
\ 11
on f(s,A} in (I). And, we decide the values of e, f, g, h so that the following transformed VTf
Then, VTf of the system (28) IS
By performing the following dynamical feedback on (28), we can let the denominator polynomial of VTF coincide with the denominator polynomial of (27),
satisfies the following two conditions CD qm(s) is (n-I) -degree polynomial. Pm(s} is a Hurwi tz polynomial.
®
(32) v=Jf (s , al· ) ul. +w • L " !f(s,a't} in (32) can be calculated by the method in (Bass et aI,
Dividing the denominator and \ II} . numerator of .-v( f s,./\. In (26) by the coefficient of s" in p (s), we m obtain A"
11
f(S,A }=
Lm,A.q .(s)'\II{, /
i=o
t.
1965) .
~A
L
A
m I • IlJ=J ,;v 'It. A 1/ m = L h; (s, al· ) A + p (s ) A . i=O"''' m (30)
(25 )
+h}/(e./\. +f}
\ 11
(s)·det{1\. ][m-lA(s,a'i)}
We decide the values of ai (i=O, I, --, n-I) so that the following three conditions are satisfied • Q) Least common multiple of denominator polynomials of all entries of If (s , aIL} is Hurwitz polynomial. @ All entries of If (s, aiL) are proper rational functions. ~ Mc Mill and e g re e 0 f If (s , al i) 1 S as low as possible.
Pi (s)
L=O .., • )..!IL • (27)
Performing the steps 1n 4.2., 4.3. on F(S,)..!/) in (27), we obtain the system corresponding to (16). 5.2. Calculation of Dynamical feedback
If{s,a,~)
designated in this way can be expressed as If(s).
The system (corresponding to (16)) obtained in 5. I., can be expressed as
5. 3. Realization by Dynamical Feedback The dynamical feedback becomes v=1f (S)IU.l, +w.
(33)
We realize If(s) in (33) with a minimal dimension, and substitute the system into (33).
from (28), we calculate backward
{
Yt""=lA1J'~1/ Ub'V"l: Ibvll I
T
[luL
i wIT(34)
v='",>i,/ [dl Vl ld7l'2.1 "uLT 1 wIT
We substitute the values of ai (i =O,I,--,n-I) obtained in 5.2 into the system corresponding to (16) in this section, and combine that sys tell- (34) .
from (29), we calculate 70
6. CONCLUSION In this paper, we proposed a method for realizing VTF, by the state-space method. By using this method, VTF can be realized with a very low dimension. 7. REFERENCES Bode, H.W. (1938). Variable equalizers. Bell Syst. Tech. J., 17, 2, pp.229-244. Newcomb, R. W. (1968). Active Integrated Circuit Synthesis. : Englewood Cliffs, NJ: PrenticeHall. Delansky, J . F. (1969). Some synthesis methods for adjustable networks using multivariable techniques. IEEE Trans. Circuits Theory, Vol.CT-16, pp.435-443. Kawakami, A., and S. Takahashi (1982a). Approximation of the characteristics of the networks containing variable elements. Trans. I.E.C.E., Japan, J65-A, 8, pp.857-863. Kawakami, A., J. Tajima, and S. Takahashi (1982b). Approximation and construction of the network containing variable elements. Electron. Commun. Japan, Vol.64A, No.4, pp . 9-18. Bass, R.W., and 1. Gura (1965). Higher-order system design via state space considerations. JACC, pp.311-318.
71
A REALIZATION METHOD OF THE TRANSFER FUNCTIONS CONTAINING VARIABLE PARAMETER ATSUSHI KAWAKAMI Kanazawa Institute of Technology, Department of Electronics, 7-1, Ougigaoka, Nonoichi-machi, Ishikawa-ken, 921 Japan Abstract. In this paper, we propose a method for realizing transfer functions containing variable parameter, by the state-space method. By using this method, variable transfer functions (VTF) can be often realized with a minimal dimension. In case that a minimal realization can not be obtained, the realization dimension can be fairly reduced. Key Words. Variable parameter; variable transfer functions; statical feedback; minimal realization; dynamical feedback
Some methods for realizing VTF have been reported (Newcomb, 1968; Delansky, 1969) and they are very interesting. But, these methods have the difficulty that the realization dimension is very high. Hence, the circuit synthesis is very complicated.
1. INTRODUCTION Circuits containing variable elements are the general name of the circuits containing elements whose values can be varied independently to time and frequency, by varing the values continuously, the designated input-output characteristics can be obtained continuously. Such variable circuits take an important place in the measurement-control system and the communication system. Especially, in the design of the variable equalizers, many researches have been done so far (Bode, 1938 et ~)
Therefore, in this paper, in order to overcome the difficulty of the above-mentioned methods, we propose a method for realizing VTF by the state-space method. In this method, we perform the statical feedback on the state-space realization form of separabledenominator VTF, so that we realize the general VTF. By using this method, VTF can be often realized with a minimal dimension.
.
Though, various methods to synthesize the circuits containing variable elements mentioned above exist, the synthesis method in terms of variable transfer functions (VTF) which consist of the complex angular frequency s and the variable parameter A is thought to be an interesting method from the point of view of the unification of the theory.
In case that minimal realization can not be obtained, we propose a method that we perform the dynamical feedback on it. By using this method, the realization dimension can be fairly reduced.
67
2. VOLTAGE TRANSFER FUNCTIONS CONTAINING VARIABLE PARAMETER Circuit functions of circuit containing m gearing variable elements, can be expressed by
4. MINIMAL REALIZATION OF VTF In (1), it is assumed that n~m. Similarly, we can realize it in case that n
(Kawakami et aI, 1982a). We call this F(s,A) variable transfer functions (VTF).
4.1. Modification of VTF We perform a bilinear transformation of A in the following
3. STATE-SPACE REALIZATION FORM OF SEPARABLE-DENOMINATOR VTF
Separable-denominator VTF in the following, can be realized with a minimal dimension (Kawakami et aI, 1982b). FS (s ,A) =g (s ,A) / {a (s) . b CA.}}
A = (cA +d) / (aA +b)
{9}
on F(s,)") 1n (1). And, we decide values of a, b, c, d so that ~m(s) of
(3)
A minimal dimension is a degree with respect to s, A, of its denominator. is (n-1)-degree polynomial and 'f'm(s) of (10) is n-degree polynomial.
FS (s , A) in (3 ) can be rea 1 i zed by the following two types of statespace form, by the method in (Kawakami ~, 1982b).
Dividing the denominator and numerator of £(s,A) in (10) by the coefficient of srt of tm(s), we obtain
Controllable-Observable (C-O) state-space realization form:
A
mN
I
\IL
\/~
J!l.N
F(S,A)=Lg·(s)1\. / Lfi(s)/\. i=o (, i=o (11 ) =
where
Ln. g. (A: ) s J. / Lrt_f j. (X) s J. . I
J.=o
d-
I
J.=o
(12 )
Y/i = (Y,--- Yml T, U1L = (u,--- umlT
(5)
Yi=>",ui
(6)
4.2. Realization of SeparableDenominator VTF Containing Unknown Quantities
(7 )
Consider the case of
Ib 2. = (
f)
(i=I,2,-----,m) 1IT.
Observable-Controllable (O-C) state-space realization form:
68
where . a~ (i=O,I,---,n-I) are unknown quant1t1es.
From (17) and (18), equation (19) can be expressed as /V
We obtain the C-O state-space • \' realizat10n form of AFS { s, ....... ,all} 1n (13). It can be expressed as
-
/
F (s , X , al·) =
[,
(20)
~/V \ li /V g • (s).I\. I { f m{s } L=o ~ L
_
1
\1
A
·fn(A:)+h(s,1\. ,al,}} .
+ CIb~ i Ib2 )T U. T
I; rL2)()[Tj/U{)T.
\/
Then, VTF of the system in (16) is
t~J =t~~~j!1 ~;~6lj!] ~~Z] ~=(e)
A
/
fm(s)fn(X)+h(s,.I\.,ali)'
(21)
We express denominator polynomial of Y(s,>(} in (II) as
0+)
,IV
_
1
/
fm(s)fn(X}+h(s,X}. 4.3. Statical Feedback Containing Unknown Quantities
(22)
Comparing (22) with (20), we calculate the values of ai (i=O, I, --, n-I) so that
We perform the following statical feedback on (14).
(23) u =If
(15)
(al't, ) ]1+ V
In case that such aj. (i=O, 1,---,n-1) don't exist, it is impossible to realize F(s,}.J in (I) with a minimal dimension, by this method. Then, we apply the realization method which we will propose 1n 5.
If (al· ) in (IS) can be ca 1 c u 1 ate d by the "method in (Bass ~, 1965). Then, the following system can be obtained.
Now, it is assumed that such a' (i=O,I,--,n-l) (equation (23) i~ satisfied) exist. Substituting those aL into (16), equation (16) is a minimal realization of Y(s,X) in (II).
<
In case that n m in F (s,>..) in (I), we consider a separabledenominator VTF in the following.
Now,
I ~/V \IL "" Fs (s ,A , al./, ) = .L g L. (s) I\. I { fm (s )
,IV
A
} = f m(s)
de t
{s En-lA II ( all.)
det
(AEm-lA 22)=fn (A)
A
(17)
£,=0 m-I
L=o
We calculate the following characteristic polynomial of (16),
A
5. I. Modification of VTF
(all.) -Ib IIf (aiL)} • de t (A.'Ern
\/
t,
5. REALIZATION METHOD BY DYNAMICAL FEEDBACK
det[-~~t1~ ~!I_(~i)j __ -~ AI?:~F}:~)J - fA21 (~L) l X'][m- IAZl -/A Z2 )+h(s,.I\. ,aiL)'
+AIm )} (24)
We obtain O-C state-space realization form of Ys(s,,(,aIL) in (24), and after that, we calculate the values of al (i=O, 1,---, m-I) through the simIlar steps.
4.4. Calculation of Unknown Quantities
JI
It,
·(La.A.
are satisfied.
=det {s~-/A
•
~
( 18)
We perform a bilinear
(19) 69
transformation of A in the following 11
A=(gA
'P
\ 11
on f(s,A} in (I). And, we decide the values of e, f, g, h so that the following transformed VTf
Then, VTf of the system (28) IS
By performing the following dynamical feedback on (28), we can let the denominator polynomial of VTF coincide with the denominator polynomial of (27),
satisfies the following two conditions CD qm(s) is (n-I) -degree polynomial. Pm(s} is a Hurwi tz polynomial.
®
(32) v=Jf (s , al· ) ul. +w • L " !f(s,a't} in (32) can be calculated by the method in (Bass et aI,
Dividing the denominator and \ II} . numerator of .-v( f s,./\. In (26) by the coefficient of s" in p (s), we m obtain A"
11
f(S,A }=
Lm,A.q .(s)'\II{, /
i=o
t.
1965) .
~A
L
A
m I • IlJ=J ,;v 'It. A 1/ m = L h; (s, al· ) A + p (s ) A . i=O"''' m (30)
(25 )
+h}/(e./\. +f}
\ 11
(s)·det{1\. ][m-lA(s,a'i)}
We decide the values of ai (i=O, I, --, n-I) so that the following three conditions are satisfied • Q) Least common multiple of denominator polynomials of all entries of If (s , aIL} is Hurwitz polynomial. @ All entries of If (s, aiL) are proper rational functions. ~ Mc Mill and e g re e 0 f If (s , al i) 1 S as low as possible.
Pi (s)
L=O .., • )..!IL • (27)
Performing the steps 1n 4.2., 4.3. on F(S,)..!/) in (27), we obtain the system corresponding to (16). 5.2. Calculation of Dynamical feedback
If{s,a,~)
designated in this way can be expressed as If(s).
The system (corresponding to (16)) obtained in 5. I., can be expressed as
5. 3. Realization by Dynamical Feedback The dynamical feedback becomes v=1f (S)IU.l, +w.
(33)
We realize If(s) in (33) with a minimal dimension, and substitute the system into (33).
from (28), we calculate backward
{
Yt""=lA1J'~1/ Ub'V"l: Ibvll I
T
[luL
i wIT(34)
v='",>i,/ [dl Vl ld7l'2.1 "uLT 1 wIT
We substitute the values of ai (i =O,I,--,n-I) obtained in 5.2 into the system corresponding to (16) in this section, and combine that sys tell- (34) .
from (29), we calculate 70
6. CONCLUSION In this paper, we proposed a method for realizing VTF, by the state-space method. By using this method, VTF can be realized with a very low dimension. 7. REFERENCES Bode, H.W. (1938). Variable equalizers. Bell Syst. Tech. J., 17, 2, pp.229-244. Newcomb, R. W. (1968). Active Integrated Circuit Synthesis. : Englewood Cliffs, NJ: PrenticeHall. Delansky, J . F. (1969). Some synthesis methods for adjustable networks using multivariable techniques. IEEE Trans. Circuits Theory, Vol.CT-16, pp.435-443. Kawakami, A., and S. Takahashi (1982a). Approximation of the characteristics of the networks containing variable elements. Trans. I.E.C.E., Japan, J65-A, 8, pp.857-863. Kawakami, A., J. Tajima, and S. Takahashi (1982b). Approximation and construction of the network containing variable elements. Electron. Commun. Japan, Vol.64A, No.4, pp . 9-18. Bass, R.W., and 1. Gura (1965). Higher-order system design via state space considerations. JACC, pp.311-318.
71