- Email: [email protected]
Construction of Diophantine Equation Corresponding to State Space Design and Doubly Coprime Factorization
2a-085
Copyrighl ~ J996 IFAC 131h Triennial World Congress. San Frnncisco. USA
CONSTRUCTION OF DIOPHANTINE EQUATION CORRESPONDING TO STATE SPACE DESIGN AND DOUBLY COPRIME FACTORIZATION Wataru Kase Department of Electrical Engineering, Osaka Institute of Technology. 5-16-1, Omiya, Asahi-ku, Osaka, 535, JAPAN. e-mail: [email protected]
Abstract: A doubly coprime factorization plays an important role in the d..ign of proper stabilizing controller for given LTI systems. In this paper, using polynomial matrix approach, a reduced-order doubly coprime factorization is considered. As a special case, this method gives the factorization which is also available using the state feedback and reduced-order observer. So the presented controller design method includes the ordinary ones and is more general. Key Words: Linear multivariable systems, Polynomial methods, Stabilizing con-
trollers, State-space formulas, State observers.
1. INTRODUCTION A douhly coprime fadorization plays an important role
in the design of proper stabilizing controller for given linear time-invariant systems especially using factorization approach (Vidyasager, 1985; Francis, 1987). The factorization is compulcd based on a state feedback and a full-order observer (Nett et al., 1984) or a reduced-order observer (Hippe, 1989; Fujimori, 1993). The direct computations are also shown (Hippe, 1988; Sugimoto and Yamamoto, 1990) when the system is gi,..,n in terms of a transfer function matrix, but a solut.ion corresponding to Hippe (\989) or Fujimori (1993) can not be obtained by these methods in general.
In this paper t using polynomial matrix approach, a reduced-order doubly coprime factorization is considered . As a special case, this method gives the factorization which is also available using the state feedback and (reduced-order) observer. So the controller design method which will be shown in this paper includes the ordinary ones (e.g., ~ett et al. , 1984; Fujimori, 1993)
and is more general. For the computations of polynomial matrices, there are some useful algorithms based on the state space parameters of given system (e.g., Wolovich, 1984; Kase et al., 1994), so the presented method can be easily applicable for the computer calculations. For the above purpose, the Diophantine equation corresponding to the design using a state feedback and a reduced-order observer in the state space is discussed. Such construction methods are shown in Hippe (1988, 1989), but these methods are roundabout. Then this result will be applied to co mpute a doubly coprime fractional representation. The relation between the presented method and the state space method using reduced-order observer (Fujimori, 1993) will be also discussed . Notations
R!' xffils) : Set of p x m polynomial matrices with real coefficients. : P x P identity matrix. BolD) : i-th column degree of polynomial matrix D. II'
1257
&r;[D] : i-th row degree of polynomial matrix D. r , [D] : The column leading coefficient matrix of D. r,[D] : The row leading coefficient matrix of D.
[~I~l
:=C(s[-A)-'B+E.
B'3 D. is column proper with &.,;[D.]
The problem disc,!!sse_d in this p"J'er !! of findingyolynomiaJ matrices D, ,~ DCl Nel DCl Ne! Hand D. for given D, N , D. and H satisfying
Note that assumption HI is not used explicitly in this paper. det D. gives the desired poles location by state feedback and det H gives the observer's poles. So if D;l Ne Nef)';1 is used as a stabilizing control1er for ND-l = fJ-1N, the assumption is necessary. It is known (Kailath , 1980) that for given D and N (or D and N) there exist polynomial matrices D, N (or D, N) , X, Y, X and Y such that
[1_XoJ[ ~ i) = l.+m X,
D, D" D., HE RpxPls], N, NE Rm Xp[s], D, DCI ii E Rm xm [s), N" N, E RP xm [B], deg det D deg d"t D. n, dog det H ~ n - rn, De} De : nonsingular
=
=
where detD. = adetO. and det H nonzero polynomials a and b.
=
H are assumed to satisfy the following assump-
BIdet D. and det H are stable polynomials, B2 D. is column proper with &dID.] = &.,;[D], B3 H is row proper with &,,[H] ~ 8,,[0]- 1. It is e"'D' to see that deg det D.
degdet H
~
X E R!' xm[s],
YE R,' xP[.],
(2)
YE Rm xm l.]
and the finding methods of such matrices which is adequate for computer calculation were reported (Beelen and Veltkamp , 1987; Kase et aI., 1994). Thus, we further assume that the polynomial ma.trices satisfying eq.(2) for D and N (or jj and N) were already obtained. With out loss of generality, we also assume that D is row proper (or D is colunm proper).
bdetH for
D and N are assumed to satisfy the following assumptions. Al D is a nonsingular matrix (without loss of generality, assume that D is column proper (Wolovich, 1974)), A2 N D-' i. proper. A3 D and N are right coprime. D. and tions.
&,,[D]- 1.
=
2. PROBLEM STATEMENT
n. ,
~
= n from B2 and
n - rn from B3 .
The dual problem is of finding polynomial matrices D , N, D el Ne} DCI Nel and H satisfying eq,(l) for given 0, N, D. and H, where deg detD. > n-p, degdet H = n, and det D. = ii det D.. det H = b det H for non.ero polynomials ii and b. 0 and N are assumed to satisfy the following assumptions . A'l jj is a nonsinguhr matrix (without loss of generality, assume that .Dis row proper (Wolovich, 1974)), A'2 jj-l it is proper, A'30 and N are left. coprime.
n.
D. and H arc assum"d to satisfy the following assumptions. Bl det D. and det H are stable polynomials, B'2 H is row proper with &.,[ii] &"ID],
3. SYNTHESIS OF DIOPHANTIl'\E EQUATION The problem considered in this section is to obtain polynomial matrices DCI Ne and H sa.tisfying Diophantine equation (3) and satisfying both H-' D, and H- I N, are proper for given D , N, D. and fI are considered. Since A3 holds, D, and N, satisfying eq.(3) exist for arbitrary H D • . If H is given , Dc and Ne are presented by
where Q E Rpxm[s] is an arbitrary polynomial matrix (Antsaklis, 1979). Properness condition for D-;' N, is presented by the following lemma. Lemma 1. Assume that AI-A3 and B2 hold in eq.(3). Then, D-; I N, is proper and det{I. + D; 1 N,N D-') '" 0 if and only if H- 1 N, is proper. Proof. From eq.(3), we obtain I DD•
= D-' e H -
=D-;IH{Ip -
I D-'N c c N D.
(5)
H-'N, ND;I).
(6)
where N D;' is proper from A2 and B2 . Let K, := tim DD;' E R!' xp and K,:= lim ND-' E R mx •.
'_00
._00
Note that K 1 is nODsingular from At and B2 and
Hm N D;:'
=
'-00
1258
= ._00 lim N D-' DD;:' =
K,K,.
(7)
If D;:1 N, is proper and detUp + D;:1 N,N D-l)
"#
0,
then Ka:= lim D;l Ne E RPxm is finite and
with de.!. D-f. and det ii being a stable polynomial. Calculate D , H E Rm xm [s] such that
.~oo
+ D; 1N,N D- I ) =
det lim (1p • ~oo
det(Jp + K 3 K,)
(8) From eq.(5) lim D;1 H = lim (DD;I
'_00
1-00
+ D;I N,N D;I)
= 1<1 + 1<31<,KI
D.xii=(jD+R,
.~oo
E RPxm is finite and
=
lim DD;I lim D;:IH(Jp - H - I N,ND;') '_00 1 - 0 (' lim D;:I H(Jp - K.K,K I ) Jp
= ._00
=
(10)
D;:IN,ND- l )
._00 + = det tim (lp + D ;: I H H- I N,N D- l )
=det[Ip + (Jp -
1<.1<,1<.)-1 K.K,Kd det(/, - K.I<,I<.)-1 I- O. 0
Corollary 1. Assume that AI-A3 and B2 hold in eq.(3) . Then, D-;I/1., is strictly proper if and only if H-l Ne is strictly proper. Assume that D;lNc is strictly proper. Then from eq.(5), Proof.
00
DD;1
= Kl
=
H- l N, D.X _ H- I Qjj D.X _ (jii- l jj D.X - fjDii- 1
=
=
= Rii- ' = sRD-l .• -1 Dii- 1 = sRD-l .• -1 jj-l jj
,-I jj-l jj is proper from B3 and eq.(14). Thus, H-I N, is proper.
' _ 00
Thus the lemma is completely proved.
(15)
For the second assertion, we will only show that the properoess of H- 1 Ne according to Lemma 1. From eq.(3),
holds for finite sand
= ,--+lim
(14)
Then, there exists a polynomial b "# 0 such that det jj = bdetH, and D;lNe is proper for polynomial matrices D, and N, given by eq.( 4).
+ D;:I N,N D-I) = detD;:I(D,D + N,N)D-I
lim D;:1 H
isstrictlyproper.
H- I Q = (jjj-l , Hand Q are len cop rime.
1-0
11--+ 00
RE
Proof. The first assertion is dear from eq.(15).
from eq.(6). The above equation implies that det(lp I<.I<,K.) "# 0 and thus D;:I H is proper. Therefore, D;:IN, = D;:IHH- I N, is proper. Finally,
=
RD- l
(j,
Then, calculate H and Q such that
Conversely, if B- 1 Ne is proper, then K,,::= lim H- 1 Ne
det lim (Ip
(t3)
where D is column proper. Next, calculate RPxm[s] such that
(9)
using eq.(7). Since [{I is nonsingular and eq.(8) holds, eq.(9) implies that both D;:I Hand H- I D, are proper. Therefore, H-INc = H-IDc ·D;lNe is proper.
det(Ip
ii f)-I = frl jj
"# O.
(nonsingular). (11)
Conversely, assume that H- l Ne is strictly proper. Then eq.(ll) holds again from eq.(6) and D;:1 H is proper. Thus D;l Ne is strictly proper. 0
Now the main theorem of this paper is stated as follow.
(16)
from eq.(13).
sRD- ' is proper from 0
Note that some useful methods for the calculation of eqs.(13)-(15) is presented in Kase et al. (1994) and Wolovich (1984). The above Theorem gives a design method of proper .tabiliz;nll controller D;: I N, with deg det D, .:; deg det if. This meth£d is interesting esspecially for the case where deg det IT n - m. A selection of if which makes the order of controller less than
=
n - m is under studying.
Now, the results of the dual problem win be stated; obtain polynomial matrices phantine equation
This implies that both D;:I Hand H- I D, are proper and H-l Ne is strictly proper.
from eq.(15) from eq.(13) from eq.(14)
Dc, Ne: and 5. satisfying Dio(17)
and both Den;· and NeD;-' are proper for given D, N, D. and if. Since A'3l)?Jgs, D,.!!I'd N, satisfying eq.(17) exist for arbitrary H D.. If D. i. determined, Dc and /Vc. are given by
Theorem 1. Assume th.t AI-A3 and BI-B3 hold . Set D. E RP xP[s] and if E Rm xm[s] as follow, (12)
where
1259
Q E Rpxm[s] i. an arbitrary polynomial matrix.
Lemma 1', in eq,(17),
Assume that A 'I-A'3 and B'2 hold Then, NJj;' is proper and detUm + D-' NNJj;') # 0 if and only if N,D;' is proper. CorolJary 1'. Assume that A'1-A'3 and B'2 hold in eq.(l1) , Then, N,D; ' is strictly proper if and only if NcD;l is strictly proper.
Theorem 1 '. Assume that A'1-A'3 and B'I-B'3 hold, Set ii E Rm xm [sJ and D. E RPX P[.] as follow 8,,[ifJ
= 8,,[DJ,
8,,[D.J?: 8,; [DJ -I
(19)
step 4 By the division algorithm, calculate the polynomial matrices Qand it satisfying D.X ii
H- 1Q = Qif- l ,
step 6 Calculate the polynomial matrices satisfying
QD; 1 = D; IQ, D.
and
D.
and
Q
Qare right coprime. (23)
(20)
where b is row proper, Next, calculate Q, such that
[D, N, I = [H D.
R E RP '''' [s]
b-'Risstrictlyproper,
(21)
Qsuch that QO;;' = D-; 'Q, D. and Qare right coprime,
(22)
D.
Hand Q are left coprime,
step 7 Set
[rIb. =D.D- I
Then, calculate
is strictly proper,
step 5 Calculate the polynomial matrices Hand Q 8atisfying
with det if and det D. being a stable polynomial. Calculate D, ff E RP XP[sJ such that
D.xif=DQ+R,
= Qfj + it, iifj-'
and
[jJ
Q
I [1 :5] ,
= [~ ~ ][
ig.].
(24)
From step 6, it is clear that there exists a polynomial
Then, there exists a polynomial a # 0 such that detD. = adetD., and both N,D;;' is proper for poly· nomial matrices 0, and N, given by eq,(18).
a
i _0 such th~ det D. = a det 5., From step = HD.Q. Thus we obtain from step 7,
5, 6,
QHD.
[D,
N,l [ ~~,]
= 0,
(25)
and thus
N~] [ I!.., N -D
4. APPLICATION TO DOUBLY COP RIME FACTORIZATION
[DN -D, N,! ] = [HD. _0_ ] 0 H D.
(26)
or equivalently Using the result of Theorem I, an algorithm to obtain a general order doubly coprjme fractional representation is presented as follows ,
Algorithm 1 .tep 1 For given D and N satisfying AI-A3, calculate the generalized Bezout identity
[NYXl[DX] -D N _Y
(27)
where DD;' and N D;' are proper from A2 and B2, and H- l D, and H- l N, are proper from Theorem 1. In the case where o.;[if] ~ ii,,[D], if-ID and if-I N are proper and thus D,}j;l and NeD;! must be proper from eq,(26). Therefore, this case includes the conventional doubly coprime factorization . In the case
= i p +m '
where ii.;[H]
step 2 Define the stable polynomial matrices D. and
H satisfying
D. is column proper with 8,; [D.J = 8,,[DJ, 8,;[ifJ ?: 8,;[0]- I.
= il.;[DJ -1 for some i, ii- I D and H- l N
are rational functions with the first differential term. This case includes the doubly coprime factorization using reduced-order observer. Although , each element in
eq,(27) is not proper, it is easy to obtain the factorization which has all proper transfer function matrices.
Thi. and its application to H 00 control problem wiU be
step 3 Calculate the polynomial malrices D and isfying
jj fJ- 1 = jj-I H,
H-'N,]
[DD;1 N'D;I] H-ID, [ ii-IN -if-ID ND;' -5,5;1 = lp+m
H.at-
reported in a forthcoming paper.
Dual algorithm to obtain a general ord er doubly coprime fractional representation will be obtained,
jj is column proper.
1260
Define Tt E R(n-m)xn , T2 E Rmxn, PE Rnx(n-m) and V E Rnx= as follows,
5. RELATION TO STATE SPACE 10 this section, the relation between the presented method and the state space method using reduced-order observer (Fujimori, 1993) will be considered, so the sit-
uation will be restricted to the case where 8,.[D.] = 8,,[D] and 8,.[H] = a,,[D.]- 1. For the case of fullorder observer (Nett et al., 1984), the same manner will be applicable. For simplicity, assume that r,[D.] r,[.o]. Let
Then,
=C,
T2
[p
= rdD] and rr[H] =
V1
TIA - FT,
[~ ] = [~ ]
=MC V 1= In
(38)
Ei{=CB+EoE.
(39)
[p
and (28)
and assume that rank C = m. From the structure theorem (Wolovich, 1974), each element of the left hand side in eq.(27) can be written aa follows
DD;-l=[~;I~;], ND;-'=[~~IB;],
BN=ME-TIB,
[-~M -~o]
and using
F = TIAP, M = TIAV, CM = -CAP, Eo = -CAV.
(40)
From eq.(36), TAT-I = eq.(31),
(29)
Thus, from eqs.(39) and (40) ,
H-1D c --
[FCc IBEy y
(31)
] 1
1 1 - [~] 5.0Ne5= [~] c. -~' . exl (f--.
--1- _
'
(32)
where I( E W" makes AK := A + BK be Hurwitz. Then using th(~ inversion and multiplication rule of transfer fun ctions (Frands, 1987) to eq.(29),
ND-l=[AC",.-_~:IB;].
(33)
Comparing eq.(28) with eq.(33) ,
DD-1_(A+BKIB] • K lp '
if-ID = H
_
D-IN=
Z-l (F Bx(C+EK)+ByK] Z _ [F 0 ] (43) o AK 0 AK ' Z-I [ By +BBx E] =
o
[=
[~ ]
(44)
,
[Co ExC+(Ey+ExE)K1Z= [Co Ey + ExE = lp,
N
[
(41)
H-ID,DD;-I + H-1N,N D;' F Bx(C+EK)+ByK By +BXE] =[ 0 AK B. C, ExC+(Ey+ExE)K Ey+ExE (42) If there exists a nonsingular matrix Z such that
ND-I=[A+BKB] • C + EK E . (34)
F M ME-B-] -CM-E-E--E-E .
F ME-T,B] '" -CAP CB-CAVE +.~.
Next, from eqs.(34) and (31),
Similarly, using the inversion and multiplication rule of transfer functions with the first differential term (Fujimori, 1993), _
N-
I-tAP -~V] +.1=,
01,
(45) (46)
(35)
E
then eq.(42) equals to Ip. Setting Z
Comparing eq.(28) with eq.(35), there exists a nonsin-
=
eq.( 43) yields,
gular matrix T E Rnxn such that
(Bx - M)C - (TIB - BxE - By)K
=0
(47)
using eq.(38). Thus, (36)
(48)
1261
Under the definition of eq.( 48), eq.( 44) holds. eqs.(4S) and (46), [C, Ex J
[~ 1+ f{ = 0
(49)
[C, Ex J
= -K [p
(50)
and
VJ
using eq.(38). Thus from eqs.(46), (48) and (50),
F TIB-ME] -KP Ip+KVE ' F M -KP -KV
J.
6. CONCLUSION
From
In this paper t the Diophantine equation corresponding to the design using a state feedback and a reduced-order observer in the state space was discussed. Then a computation of a doubly eoprime fractional representation W8S given _ The dual results and the relation between
Ihe presented method and the state space method UBing reduced-order observer (Fujimori, 1993) were also discussed .
(51) REFERENCES Antsaklis, P.J.(1979). Some relations satisfied by prime polynomial matrices and their role in linear mul-
Finally, from eqs.(4J) and (32),
tivariable system theory. IEEE 1rans. Automat. Contr., AC-24, pp.611-616. Beelen, Th.G.J. and G.W . Veltkamp (1987). Numerical
jj-I DD)j;l + jj-I if N,D;I F MCy + (M E-TI B)Cx = 0 AK B" [ - CA P CBCx-CAVCK +CKA CKB, K CK ECx+q::·
0]
=
computation of a coptime factorization of a transfer function matrix. System<; and Control Letters, 9,
(52)
If there exists a nonsingular matrix Z such that
Z-I
[Fo MCy +(ME-TIB)C x ] Z= [F 0 ] ,(53) AK 0 AK
Z-I
[~J = [~J ,
(54)
[-CAP CBCx - CAVCK = [ - CAP 0 J ' CK B , = Im,
+ CKAK J Z
then eq.(S2) equals to Im . ln [ ~], eq.(S3) yields
(55) (56)
=
Again, setting Z
om
M(ECx
+ Cl' -
C) - TIB(K
x
Cl'
= C+ EK,
+ Cx ) = 0 CK
= C.
Under the definition of eq.(58), eq.(S5) holds. eqs.(S4), (S6) and (58) ,
B, = V
(57)
(58) From
(59)
using eq.(38). Thus from eqs.(S8) and (59), -
--I
D,D.
compensators by using reduced·order observers. IEEE 1rans. Automat. Contr., 38, pp.1435-1439. Hippe, P.(1988). Parameterization of the full-order compensator in the frequency domain. Int. J. Control, 48 , pp.IS83-1603. Hippe, P.( 1989). Modified doubly coprime fractional representations related to the red uced-order observer. IEEE 1hns . Automat. Contr., 34, pp.573575. Kailath , T.(1980). Linear systems, Prentice-Hall, NJ. Kase, W ., K. Tamura and P. N. Nikiforuk (1994). A calculation of minimal bf\Sis in polynomial vector
using eq.(38) . Thus ,
C = -K,
pp.281-288. Frauds, B.A.(1987). A course in H~ control theory. Lecture Notes in Control and Informa.tion Sciences, 88 , Springer-Verlag, Berlin. Fujimori, A.(1993). Parameterizations of stabilizing
AK IV] - - - I [~] = [ C+EKO ' N,D. = -=KfO . (60)
Eqs .(34), (41), (SI) and (60) are equivalent to eq.(17a) given in Fujimori (1993).
space. in Systems and Nettworks : Ma,hematicaJ Theory and Applications, Vol.Il, Mathematical Research , 79, U. Helmke and R. Mennicken (Eds.), pp.275-278, Akademie Verlag, Berlin. Nett, C. N., C.A. Jacobson and M. J. Balas (1984). A connection between state-space and doubly c<>prime fractional representation. IEEE 'Irans. Automat. Contr., 29 , pp.831-832 . Sugimoto, K. and Y. Yamamoto (1990). A polynomial matrix method for computing stable rational doubly coprime factorizations. Systems and Control Letters, 14, pp.267-273. Vidyasager, M.(1985) . Control system synthesis: a factorization approach, MIT Press, MA . Wolovich, W.A.(1974). Linear multi variable systems, Springer-Verlag, Berlin. Woloviob, W.A.(I984). A division algorithm for polynomial matrices. IEEE 1rans. Automat. Contr., 29, pp .656-658.
1262
Copyrighl ~ J996 IFAC 131h Triennial World Congress. San Frnncisco. USA
CONSTRUCTION OF DIOPHANTINE EQUATION CORRESPONDING TO STATE SPACE DESIGN AND DOUBLY COPRIME FACTORIZATION Wataru Kase Department of Electrical Engineering, Osaka Institute of Technology. 5-16-1, Omiya, Asahi-ku, Osaka, 535, JAPAN. e-mail: [email protected]
Abstract: A doubly coprime factorization plays an important role in the d..ign of proper stabilizing controller for given LTI systems. In this paper, using polynomial matrix approach, a reduced-order doubly coprime factorization is considered. As a special case, this method gives the factorization which is also available using the state feedback and reduced-order observer. So the presented controller design method includes the ordinary ones and is more general. Key Words: Linear multivariable systems, Polynomial methods, Stabilizing con-
trollers, State-space formulas, State observers.
1. INTRODUCTION A douhly coprime fadorization plays an important role
in the design of proper stabilizing controller for given linear time-invariant systems especially using factorization approach (Vidyasager, 1985; Francis, 1987). The factorization is compulcd based on a state feedback and a full-order observer (Nett et al., 1984) or a reduced-order observer (Hippe, 1989; Fujimori, 1993). The direct computations are also shown (Hippe, 1988; Sugimoto and Yamamoto, 1990) when the system is gi,..,n in terms of a transfer function matrix, but a solut.ion corresponding to Hippe (\989) or Fujimori (1993) can not be obtained by these methods in general.
In this paper t using polynomial matrix approach, a reduced-order doubly coprime factorization is considered . As a special case, this method gives the factorization which is also available using the state feedback and (reduced-order) observer. So the controller design method which will be shown in this paper includes the ordinary ones (e.g., ~ett et al. , 1984; Fujimori, 1993)
and is more general. For the computations of polynomial matrices, there are some useful algorithms based on the state space parameters of given system (e.g., Wolovich, 1984; Kase et al., 1994), so the presented method can be easily applicable for the computer calculations. For the above purpose, the Diophantine equation corresponding to the design using a state feedback and a reduced-order observer in the state space is discussed. Such construction methods are shown in Hippe (1988, 1989), but these methods are roundabout. Then this result will be applied to co mpute a doubly coprime fractional representation. The relation between the presented method and the state space method using reduced-order observer (Fujimori, 1993) will be also discussed . Notations
R!' xffils) : Set of p x m polynomial matrices with real coefficients. : P x P identity matrix. BolD) : i-th column degree of polynomial matrix D. II'
1257
&r;[D] : i-th row degree of polynomial matrix D. r , [D] : The column leading coefficient matrix of D. r,[D] : The row leading coefficient matrix of D.
[~I~l
:=C(s[-A)-'B+E.
B'3 D. is column proper with &.,;[D.]
The problem disc,!!sse_d in this p"J'er !! of findingyolynomiaJ matrices D, ,~ DCl Nel DCl Ne! Hand D. for given D, N , D. and H satisfying
Note that assumption HI is not used explicitly in this paper. det D. gives the desired poles location by state feedback and det H gives the observer's poles. So if D;l Ne Nef)';1 is used as a stabilizing control1er for ND-l = fJ-1N, the assumption is necessary. It is known (Kailath , 1980) that for given D and N (or D and N) there exist polynomial matrices D, N (or D, N) , X, Y, X and Y such that
[1_XoJ[ ~ i) = l.+m X,
D, D" D., HE RpxPls], N, NE Rm Xp[s], D, DCI ii E Rm xm [s), N" N, E RP xm [B], deg det D deg d"t D. n, dog det H ~ n - rn, De} De : nonsingular
=
=
where detD. = adetO. and det H nonzero polynomials a and b.
=
H are assumed to satisfy the following assump-
BIdet D. and det H are stable polynomials, B2 D. is column proper with &dID.] = &.,;[D], B3 H is row proper with &,,[H] ~ 8,,[0]- 1. It is e"'D' to see that deg det D.
degdet H
~
X E R!' xm[s],
YE R,' xP[.],
(2)
YE Rm xm l.]
and the finding methods of such matrices which is adequate for computer calculation were reported (Beelen and Veltkamp , 1987; Kase et aI., 1994). Thus, we further assume that the polynomial ma.trices satisfying eq.(2) for D and N (or jj and N) were already obtained. With out loss of generality, we also assume that D is row proper (or D is colunm proper).
bdetH for
D and N are assumed to satisfy the following assumptions. Al D is a nonsingular matrix (without loss of generality, assume that D is column proper (Wolovich, 1974)), A2 N D-' i. proper. A3 D and N are right coprime. D. and tions.
&,,[D]- 1.
=
2. PROBLEM STATEMENT
n. ,
~
= n from B2 and
n - rn from B3 .
The dual problem is of finding polynomial matrices D , N, D el Ne} DCI Nel and H satisfying eq,(l) for given 0, N, D. and H, where deg detD. > n-p, degdet H = n, and det D. = ii det D.. det H = b det H for non.ero polynomials ii and b. 0 and N are assumed to satisfy the following assumptions . A'l jj is a nonsinguhr matrix (without loss of generality, assume that .Dis row proper (Wolovich, 1974)), A'2 jj-l it is proper, A'30 and N are left. coprime.
n.
D. and H arc assum"d to satisfy the following assumptions. Bl det D. and det H are stable polynomials, B'2 H is row proper with &.,[ii] &"ID],
3. SYNTHESIS OF DIOPHANTIl'\E EQUATION The problem considered in this section is to obtain polynomial matrices DCI Ne and H sa.tisfying Diophantine equation (3) and satisfying both H-' D, and H- I N, are proper for given D , N, D. and fI are considered. Since A3 holds, D, and N, satisfying eq.(3) exist for arbitrary H D • . If H is given , Dc and Ne are presented by
where Q E Rpxm[s] is an arbitrary polynomial matrix (Antsaklis, 1979). Properness condition for D-;' N, is presented by the following lemma. Lemma 1. Assume that AI-A3 and B2 hold in eq.(3). Then, D-; I N, is proper and det{I. + D; 1 N,N D-') '" 0 if and only if H- 1 N, is proper. Proof. From eq.(3), we obtain I DD•
= D-' e H -
=D-;IH{Ip -
I D-'N c c N D.
(5)
H-'N, ND;I).
(6)
where N D;' is proper from A2 and B2 . Let K, := tim DD;' E R!' xp and K,:= lim ND-' E R mx •.
'_00
._00
Note that K 1 is nODsingular from At and B2 and
Hm N D;:'
=
'-00
1258
= ._00 lim N D-' DD;:' =
K,K,.
(7)
If D;:1 N, is proper and detUp + D;:1 N,N D-l)
"#
0,
then Ka:= lim D;l Ne E RPxm is finite and
with de.!. D-f. and det ii being a stable polynomial. Calculate D , H E Rm xm [s] such that
.~oo
+ D; 1N,N D- I ) =
det lim (1p • ~oo
det(Jp + K 3 K,)
(8) From eq.(5) lim D;1 H = lim (DD;I
'_00
1-00
+ D;I N,N D;I)
= 1<1 + 1<31<,KI
D.xii=(jD+R,
.~oo
E RPxm is finite and
=
lim DD;I lim D;:IH(Jp - H - I N,ND;') '_00 1 - 0 (' lim D;:I H(Jp - K.K,K I ) Jp
= ._00
=
(10)
D;:IN,ND- l )
._00 + = det tim (lp + D ;: I H H- I N,N D- l )
=det[Ip + (Jp -
1<.1<,1<.)-1 K.K,Kd det(/, - K.I<,I<.)-1 I- O. 0
Corollary 1. Assume that AI-A3 and B2 hold in eq.(3) . Then, D-;I/1., is strictly proper if and only if H-l Ne is strictly proper. Assume that D;lNc is strictly proper. Then from eq.(5), Proof.
00
DD;1
= Kl
=
H- l N, D.X _ H- I Qjj D.X _ (jii- l jj D.X - fjDii- 1
=
=
= Rii- ' = sRD-l .• -1 Dii- 1 = sRD-l .• -1 jj-l jj
,-I jj-l jj is proper from B3 and eq.(14). Thus, H-I N, is proper.
' _ 00
Thus the lemma is completely proved.
(15)
For the second assertion, we will only show that the properoess of H- 1 Ne according to Lemma 1. From eq.(3),
holds for finite sand
= ,--+lim
(14)
Then, there exists a polynomial b "# 0 such that det jj = bdetH, and D;lNe is proper for polynomial matrices D, and N, given by eq.( 4).
+ D;:I N,N D-I) = detD;:I(D,D + N,N)D-I
lim D;:1 H
isstrictlyproper.
H- I Q = (jjj-l , Hand Q are len cop rime.
1-0
11--+ 00
RE
Proof. The first assertion is dear from eq.(15).
from eq.(6). The above equation implies that det(lp I<.I<,K.) "# 0 and thus D;:I H is proper. Therefore, D;:IN, = D;:IHH- I N, is proper. Finally,
=
RD- l
(j,
Then, calculate H and Q such that
Conversely, if B- 1 Ne is proper, then K,,::= lim H- 1 Ne
det lim (Ip
(t3)
where D is column proper. Next, calculate RPxm[s] such that
(9)
using eq.(7). Since [{I is nonsingular and eq.(8) holds, eq.(9) implies that both D;:I Hand H- I D, are proper. Therefore, H-INc = H-IDc ·D;lNe is proper.
det(Ip
ii f)-I = frl jj
"# O.
(nonsingular). (11)
Conversely, assume that H- l Ne is strictly proper. Then eq.(ll) holds again from eq.(6) and D;:1 H is proper. Thus D;l Ne is strictly proper. 0
Now the main theorem of this paper is stated as follow.
(16)
from eq.(13).
sRD- ' is proper from 0
Note that some useful methods for the calculation of eqs.(13)-(15) is presented in Kase et al. (1994) and Wolovich (1984). The above Theorem gives a design method of proper .tabiliz;nll controller D;: I N, with deg det D, .:; deg det if. This meth£d is interesting esspecially for the case where deg det IT n - m. A selection of if which makes the order of controller less than
=
n - m is under studying.
Now, the results of the dual problem win be stated; obtain polynomial matrices phantine equation
This implies that both D;:I Hand H- I D, are proper and H-l Ne is strictly proper.
from eq.(15) from eq.(13) from eq.(14)
Dc, Ne: and 5. satisfying Dio(17)
and both Den;· and NeD;-' are proper for given D, N, D. and if. Since A'3l)?Jgs, D,.!!I'd N, satisfying eq.(17) exist for arbitrary H D.. If D. i. determined, Dc and /Vc. are given by
Theorem 1. Assume th.t AI-A3 and BI-B3 hold . Set D. E RP xP[s] and if E Rm xm[s] as follow, (12)
where
1259
Q E Rpxm[s] i. an arbitrary polynomial matrix.
Lemma 1', in eq,(17),
Assume that A 'I-A'3 and B'2 hold Then, NJj;' is proper and detUm + D-' NNJj;') # 0 if and only if N,D;' is proper. CorolJary 1'. Assume that A'1-A'3 and B'2 hold in eq.(l1) , Then, N,D; ' is strictly proper if and only if NcD;l is strictly proper.
Theorem 1 '. Assume that A'1-A'3 and B'I-B'3 hold, Set ii E Rm xm [sJ and D. E RPX P[.] as follow 8,,[ifJ
= 8,,[DJ,
8,,[D.J?: 8,; [DJ -I
(19)
step 4 By the division algorithm, calculate the polynomial matrices Qand it satisfying D.X ii
H- 1Q = Qif- l ,
step 6 Calculate the polynomial matrices satisfying
QD; 1 = D; IQ, D.
and
D.
and
Q
Qare right coprime. (23)
(20)
where b is row proper, Next, calculate Q, such that
[D, N, I = [H D.
R E RP '''' [s]
b-'Risstrictlyproper,
(21)
Qsuch that QO;;' = D-; 'Q, D. and Qare right coprime,
(22)
D.
Hand Q are left coprime,
step 7 Set
[rIb. =D.D- I
Then, calculate
is strictly proper,
step 5 Calculate the polynomial matrices Hand Q 8atisfying
with det if and det D. being a stable polynomial. Calculate D, ff E RP XP[sJ such that
D.xif=DQ+R,
= Qfj + it, iifj-'
and
[jJ
Q
I [1 :5] ,
= [~ ~ ][
ig.].
(24)
From step 6, it is clear that there exists a polynomial
Then, there exists a polynomial a # 0 such that detD. = adetD., and both N,D;;' is proper for poly· nomial matrices 0, and N, given by eq,(18).
a
i _0 such th~ det D. = a det 5., From step = HD.Q. Thus we obtain from step 7,
5, 6,
QHD.
[D,
N,l [ ~~,]
= 0,
(25)
and thus
N~] [ I!.., N -D
4. APPLICATION TO DOUBLY COP RIME FACTORIZATION
[DN -D, N,! ] = [HD. _0_ ] 0 H D.
(26)
or equivalently Using the result of Theorem I, an algorithm to obtain a general order doubly coprjme fractional representation is presented as follows ,
Algorithm 1 .tep 1 For given D and N satisfying AI-A3, calculate the generalized Bezout identity
[NYXl[DX] -D N _Y
(27)
where DD;' and N D;' are proper from A2 and B2, and H- l D, and H- l N, are proper from Theorem 1. In the case where o.;[if] ~ ii,,[D], if-ID and if-I N are proper and thus D,}j;l and NeD;! must be proper from eq,(26). Therefore, this case includes the conventional doubly coprime factorization . In the case
= i p +m '
where ii.;[H]
step 2 Define the stable polynomial matrices D. and
H satisfying
D. is column proper with 8,; [D.J = 8,,[DJ, 8,;[ifJ ?: 8,;[0]- I.
= il.;[DJ -1 for some i, ii- I D and H- l N
are rational functions with the first differential term. This case includes the doubly coprime factorization using reduced-order observer. Although , each element in
eq,(27) is not proper, it is easy to obtain the factorization which has all proper transfer function matrices.
Thi. and its application to H 00 control problem wiU be
step 3 Calculate the polynomial malrices D and isfying
jj fJ- 1 = jj-I H,
H-'N,]
[DD;1 N'D;I] H-ID, [ ii-IN -if-ID ND;' -5,5;1 = lp+m
H.at-
reported in a forthcoming paper.
Dual algorithm to obtain a general ord er doubly coprime fractional representation will be obtained,
jj is column proper.
1260
Define Tt E R(n-m)xn , T2 E Rmxn, PE Rnx(n-m) and V E Rnx= as follows,
5. RELATION TO STATE SPACE 10 this section, the relation between the presented method and the state space method using reduced-order observer (Fujimori, 1993) will be considered, so the sit-
uation will be restricted to the case where 8,.[D.] = 8,,[D] and 8,.[H] = a,,[D.]- 1. For the case of fullorder observer (Nett et al., 1984), the same manner will be applicable. For simplicity, assume that r,[D.] r,[.o]. Let
Then,
=C,
T2
[p
= rdD] and rr[H] =
V1
TIA - FT,
[~ ] = [~ ]
=MC V 1= In
(38)
Ei{=CB+EoE.
(39)
[p
and (28)
and assume that rank C = m. From the structure theorem (Wolovich, 1974), each element of the left hand side in eq.(27) can be written aa follows
DD;-l=[~;I~;], ND;-'=[~~IB;],
BN=ME-TIB,
[-~M -~o]
and using
F = TIAP, M = TIAV, CM = -CAP, Eo = -CAV.
(40)
From eq.(36), TAT-I = eq.(31),
(29)
Thus, from eqs.(39) and (40) ,
H-1D c --
[FCc IBEy y
(31)
] 1
1 1 - [~] 5.0Ne5= [~] c. -~' . exl (f--.
--1- _
'
(32)
where I( E W" makes AK := A + BK be Hurwitz. Then using th(~ inversion and multiplication rule of transfer fun ctions (Frands, 1987) to eq.(29),
ND-l=[AC",.-_~:IB;].
(33)
Comparing eq.(28) with eq.(33) ,
DD-1_(A+BKIB] • K lp '
if-ID = H
_
D-IN=
Z-l (F Bx(C+EK)+ByK] Z _ [F 0 ] (43) o AK 0 AK ' Z-I [ By +BBx E] =
o
[=
[~ ]
(44)
,
[Co ExC+(Ey+ExE)K1Z= [Co Ey + ExE = lp,
N
[
(41)
H-ID,DD;-I + H-1N,N D;' F Bx(C+EK)+ByK By +BXE] =[ 0 AK B. C, ExC+(Ey+ExE)K Ey+ExE (42) If there exists a nonsingular matrix Z such that
ND-I=[A+BKB] • C + EK E . (34)
F M ME-B-] -CM-E-E--E-E .
F ME-T,B] '" -CAP CB-CAVE +.~.
Next, from eqs.(34) and (31),
Similarly, using the inversion and multiplication rule of transfer functions with the first differential term (Fujimori, 1993), _
N-
I-tAP -~V] +.1=,
01,
(45) (46)
(35)
E
then eq.(42) equals to Ip. Setting Z
Comparing eq.(28) with eq.(35), there exists a nonsin-
=
eq.( 43) yields,
gular matrix T E Rnxn such that
(Bx - M)C - (TIB - BxE - By)K
=0
(47)
using eq.(38). Thus, (36)
(48)
1261
Under the definition of eq.( 48), eq.( 44) holds. eqs.(4S) and (46), [C, Ex J
[~ 1+ f{ = 0
(49)
[C, Ex J
= -K [p
(50)
and
VJ
using eq.(38). Thus from eqs.(46), (48) and (50),
F TIB-ME] -KP Ip+KVE ' F M -KP -KV
J.
6. CONCLUSION
From
In this paper t the Diophantine equation corresponding to the design using a state feedback and a reduced-order observer in the state space was discussed. Then a computation of a doubly eoprime fractional representation W8S given _ The dual results and the relation between
Ihe presented method and the state space method UBing reduced-order observer (Fujimori, 1993) were also discussed .
(51) REFERENCES Antsaklis, P.J.(1979). Some relations satisfied by prime polynomial matrices and their role in linear mul-
Finally, from eqs.(4J) and (32),
tivariable system theory. IEEE 1rans. Automat. Contr., AC-24, pp.611-616. Beelen, Th.G.J. and G.W . Veltkamp (1987). Numerical
jj-I DD)j;l + jj-I if N,D;I F MCy + (M E-TI B)Cx = 0 AK B" [ - CA P CBCx-CAVCK +CKA CKB, K CK ECx+q::·
0]
=
computation of a coptime factorization of a transfer function matrix. System<; and Control Letters, 9,
(52)
If there exists a nonsingular matrix Z such that
Z-I
[Fo MCy +(ME-TIB)C x ] Z= [F 0 ] ,(53) AK 0 AK
Z-I
[~J = [~J ,
(54)
[-CAP CBCx - CAVCK = [ - CAP 0 J ' CK B , = Im,
+ CKAK J Z
then eq.(S2) equals to Im . ln [ ~], eq.(S3) yields
(55) (56)
=
Again, setting Z
om
M(ECx
+ Cl' -
C) - TIB(K
x
Cl'
= C+ EK,
+ Cx ) = 0 CK
= C.
Under the definition of eq.(58), eq.(S5) holds. eqs.(S4), (S6) and (58) ,
B, = V
(57)
(58) From
(59)
using eq.(38). Thus from eqs.(S8) and (59), -
--I
D,D.
compensators by using reduced·order observers. IEEE 1rans. Automat. Contr., 38, pp.1435-1439. Hippe, P.(1988). Parameterization of the full-order compensator in the frequency domain. Int. J. Control, 48 , pp.IS83-1603. Hippe, P.( 1989). Modified doubly coprime fractional representations related to the red uced-order observer. IEEE 1hns . Automat. Contr., 34, pp.573575. Kailath , T.(1980). Linear systems, Prentice-Hall, NJ. Kase, W ., K. Tamura and P. N. Nikiforuk (1994). A calculation of minimal bf\Sis in polynomial vector
using eq.(38) . Thus ,
C = -K,
pp.281-288. Frauds, B.A.(1987). A course in H~ control theory. Lecture Notes in Control and Informa.tion Sciences, 88 , Springer-Verlag, Berlin. Fujimori, A.(1993). Parameterizations of stabilizing
AK IV] - - - I [~] = [ C+EKO ' N,D. = -=KfO . (60)
Eqs .(34), (41), (SI) and (60) are equivalent to eq.(17a) given in Fujimori (1993).
space. in Systems and Nettworks : Ma,hematicaJ Theory and Applications, Vol.Il, Mathematical Research , 79, U. Helmke and R. Mennicken (Eds.), pp.275-278, Akademie Verlag, Berlin. Nett, C. N., C.A. Jacobson and M. J. Balas (1984). A connection between state-space and doubly c<>prime fractional representation. IEEE 'Irans. Automat. Contr., 29 , pp.831-832 . Sugimoto, K. and Y. Yamamoto (1990). A polynomial matrix method for computing stable rational doubly coprime factorizations. Systems and Control Letters, 14, pp.267-273. Vidyasager, M.(1985) . Control system synthesis: a factorization approach, MIT Press, MA . Wolovich, W.A.(1974). Linear multi variable systems, Springer-Verlag, Berlin. Woloviob, W.A.(I984). A division algorithm for polynomial matrices. IEEE 1rans. Automat. Contr., 29, pp .656-658.
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