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Minimal order realization with a special coordinate for matrix fraction descriptions
Minimal
Order Realization
Coordinate for
Matrix
with a Special
Fraction
Descriptions”
by w. s. CHEN and J.S.H.TSAI Control System Laboratory, Dqurtnzent ef Electrical Engineering, Nutionul Cheng Kung Unioersity, Tuinan 70101, Taiwan, Republic
ABSTRACT
structively
This paper presents
:
determining generalized
pseudoproper description is prored
state-space
or a column(roMl)-proper (MFD).
of the system
giren right(left)
and minimal realization representation from
or a nonstrictly-proper
The realized state-space
to be controlluhle
the> dimension
I.
a generalized
and observable
represent&ion
.form
algorithm for
a so-called rQht(lefi)
con-
column (row)matri.x Jiaction
with a special coordinute
in the sense of Rosenbrock
aficr reulizution
qf China
and Cobb;
is equul to the determinantal
degree
besides, of the
MFD in the normal sense.
Introduction Consider
an nth order linear, time-invariant Ei(t)
system characterized
= Ax(t) +Bu(t)
by (la)
v(t) = Wt),
(lb)
where x(t) E R”, u(t) E R”‘, y(t) E Rp, and E, A, B and C are real constant matrices with appropriate dimensions, and E is possibly singular. If the matrix E in system (1) is regular, then the system (1) is called a “regular system”. Otherwise, system (1) is called “singular”, or “generalized state space”, or “descriptor”. or “semistate”. Singular systems have received much attention due to extensive applications in many areas (14). However, some system models may not appear in the above form ; for example, the system realized by Ei(t)
= Ax(t) + B+(t)
+ B,u’“(t)
y(t) = Cx(t)+D”u(t)+D,u”‘(t)+
(24 ... +D,u’“‘(t)
(2b)
is not in the form of(l), where x(t) E R” is the state, u(t) E R”’ is the feasible control input, d”(t) denotes the set of i times continuous differentiable functions of u(t), y(t) E R”is the measure output. Here it is assumed that u(t) is sufficiently continuous differentiable. The main feature of such systems is that their input and output *This work was supported by the National Science Council of the Republic of China under contract NSC-8 1-0404-E-006-572. 815
W. S. Chen and J. S. H. Tsai include not only the function u(t), as for classical linear systems, but also its derivatives. In this paper, we propose a generalized, minimal and constructive realization algorithm that determines a controllable and observable generalized state-space system of the form shown in (2) from a column(row)-proper, a column(row)pseudoproper and a nonstrictly-proper rational transfer matrix T(s) in right(left) MFD T(s) = N,(s)&
‘(s)( =D,- ‘(s)N,(s)),
(3a)
respectively. If T(s) is column(row) proper, the realized matrix E is an identity matrix with an appropriate dimension. However, when the given MFD is column(row) pseudoproper or nonstrictly proper, E will be a singular matrix with a special coordinate form by the realization algorithm proposed in this paper. Bender and Laub (5), Lin and Yang (6) and Wang et al. (7) all decomposed the matrix E into a special form Ii
0
E= [ 0
0
1
(3b)
for the goal of system analysis and design, where I, denotes the,ix,j identity matrix andi = rank (E). By such a decomposition, the finite and infinite properties of a singular system, such as controllability, observability and stability of feedback control become more transparent and easier to understand. When T(s) is column(row) pseudoproper, the realized matrix Eobtained by the algorithm proposed in this paper is in the special coordinate form as in (3b), and the realized pair controller(quasi-right-type observer) {A, B) ({A, C)) forms a quasi-bottom-type canonical form, which has been precisely defined by Tsai and Chen (8). As for the nonstrictly proper MFD, the state-space representation in (2) is considered in this paper. The main purpose of this realization algorithm is to put more emphasis on preserving the structure characteristic of the given right(left) MFD at finite and infinite modes. In practical situations, when D,(.s)(D,(s)) is column(row) reduced, the MFD has no zero structures at infinity. However, the system (I) may result from physical modelling or from some kind of equivalent transformation, then the column(row) reduction is seldom met. Of course, the strict part can be separated by applying the division theorem for polynomial matrices, or one can obtain the column(row) reduction by employing unimodular matrices. If it is done the structural properties of the system (1) are completely altered at infinity. Therefore, there are very interesting practical applications that provide important ground for studying the realizable state-space representation of the given MFD. Recently, many algorithms have been proposed in this topic. Tan and Vandewalle (9, 10) proposed two algorithms that realize singular systems to be strongly controllable, but the realized systems are not guaranteed to be strongly observable or have any special form of E. The same deficiency appears in Shiotssuki and Kawaji’s work (11). Fang and Chang (12) have successfully improved upon the above disadvantages. However, the generalized state-space form for a given column(row)-
Matrix
Fraction Descriptions
proper,case obtained there is not in the controller(observer) canonical form. An alternative constructive algorithm has also been presented by Fang and Chang (13) for the realization of descriptor systems. In their work, as a nonstrictly-proper system is represented in the so-called pseudo-normalized left matrix fraction description (PNLMFD) form, the realized descriptor system is guaranteed to be strongly observable and controllable (strongly irreducible). However, the polynomial matrices N,(s) and D,(s) need to satisfy the following conditions: N,(s)] is row reduced. Note (i) D,(s) and N,(s) are left coprime and (ii) [D,(s) that T(s) = D, ‘(s)N,(s) may be a nonstrictly-proper rational matrix. Besides, for a T(s) = DIP’(s)N,(s), if the row degree of N,(s) is equal to the row degree of [D,(s) N,(s)] and N,(s) is also row reduced, then their algorithm (13) is a minimal realization algorithm. Note that the case of pseudo-normalized right MFD has not been discussed in their paper. Tsai and Chen (8) have successfully developed a minimal realization algorithm without satisfying the above condition (ii), [D,(s) N,(s)] is row reduced. However, in our previous work (8) the algorithm cannot be directly applied to a nonstrictlyproper MFD if it is not in the column(row)-pseudoproper form. In this paper, the minimal realization of T(s) = Dc’(s)N,(s) (=N,(s)D,‘(s)) can be obtained with the condition of left(right)-coprime row(column)-pseudoproper or nonstrictlyN,(s)] ([IV: D:]‘) is not row(column) proper rational matrix only, even if [D,(s) reduced. Moreover, if T(S) is column(row)-proper, then the generalized state-space form {A, B}({A, C}) obtained in this paper is in the standard bottom-type(righttype) controller(observer) canonical form, and the matrix E is an identity matrix with an appropriate dimension. Compared to the existing ones, ours are much more constructive and without loss of generality. Also, the realized system with a special coordinate form is guaranteed to be strongly controllable and observable if the MFD is right(left) coprime in column(row)-pseudoproper or column(row)-proper or nonstrictlyproper form. According to the definition of Cobb (14) and in the sense of Rosenbrock (15) and Dai (16), the controllability and observability conditions can be easily derived by using these structures and they become more easily checked by the core realization proposed in this paper. II. Preliminaries Firstly,
let us review some properties
of the MFD.
Dqfinition 2.1
If a square polynomial matrix R(s) is the common right(left) divisor and it is also the left(right) multipler of each common right(left) divisor of N(s) and D(s), then R(s) is the greatest common right(left) divisor of N(s) and D(s). If R(s) is a unimodular matrix, then N(s) and D(s) are right(left) coprime. DeJinition 2.2 (8)
Consider
a nonsingular
polynomial
matrix D(s) with a dimension
K,, = the highest degree of the ith column
of D(s)
p x p, and let
W. S. Chen and J. S. H. Tsai 4,
= the highest degree of the ith row of D(s).
If degdet then D(s) is column
reduced.
D(s) = c ICY), *=I
If degdet
D(s) = f; tirir i= I
then D(s) is row reduced. D
of the form ’+
+d(,,,,
(4)
where each coefficient d,, (j = 0, 1, , I?) belongs to a real field. Here d,, may not be required to be nonzero and q will be called the generalized degree of the polynomial in (4) in order to distinguish it from the normal definition of polynomial degree. To construct a p x II? polynomial matrix in a so-called column-based form, they choose p.~ polynomials from Fk, [s] and arrange them into the first column such that [d, (s)dz(s) .
d,,(s)]’ = [doIdol. . . do,,]‘.+ + [d, ,d,,.
d,,,]‘.rkl ’
+‘..+[d,,,d~,12...d,,,,1’,
(5a)
where the superscript “t” denotes transpose. As in (5a), one again chooses ps polynomials from FA1[s] and arranges them into the second column, carrying out these procedures until nz columns are entirely constructed. The polynomial matrix constructed in this way is said to be in column-based form. The indices k,, which may be all different, denote the ith generalized column degree of the polynomial matrix in column-based form. The matrix polynomial, already used widely in the literature (17, 18) whose forms are normally described as D(s) = D,,+D,.s+Dzs’+
... +D,,f,
(5b)
where D(s) E R[s]“““‘, D, E RPX”’ may be viewed as a special case of the polynomial matrix in column-based form where the generalized column degrees are all the same. They also call the degree of the determinant of a polynomial matrix, in column-based form, the generalized determinantal degree. It is equal to the sum of all generalized column degrees. Assume the given right MFD (3a) is already in column-based form, then it will be called column pseudoproper if k, > I’, (i = 1, 2, . . . . m) where k, and Y, are the ith generalized column degrees of D(s) and N(s), respectively. Similarly, these definitions can be appropriately extended to those of the row-pseudoproper cases.
Matrix
Fraction Descriptions
Definition 2.4 (8) In Definition 2.3, if the term D(s) is column reduced, then the given right MFD (3a) is called a column-proper right MFD. Similarly, this definition can be appropriately extended to the row-proper case. Following Tan and Vandewalle (9, lo), we assume the minimal value of ki is zero in this paper. Note that polynomials d,(s) = 0 - +s+ 1 E F2[s] and d2(.s) = s+ 1 E F, [s] from two different sets are not the same. Naturally, a nonzero scalar dgF,,[s] should not be confused with the polynomial d,(s) = O~syO*sy -‘f ... + 0 - s + d which is an element of F,[s]. These definitions ensure algebraically that the operations involved do not destroy system behaviour either at finite or infinite frequencies.
s*
Lemma 2.1 (14, 15) Consider the system (1). (i) It is controllable for all s. (ii) It is observable rank for all s.
B] and [E
if and only if both [&-A if and only if both
B] are full rank
C’]’ and [E’
[(X-A)’
Ctlt are full
Lemma 2.2 (16) The system i(t)
= Ax(t)+B,u(t)+B,ti(t)
(64
(6b)
y(t) = C-x(t) is state controllable if and only if %![B,,+AB,, x(t) E R” and B?(o) represents the range space of
A(B,+AB,), l
B,] = R”, where
.
III. Main Results Dqfinition 3.1 Define k,, = the ith column
degree of [Or(s)
Ur(s) = block diag([,+]},i= V:(s) = block diag ([ls..
N,(s)], i = 1,2,
. ,m
1,2,...,m .,+-‘]},i
= 1,2,. . . ,m,
and such that D, (7) = L&Ou,(s) + LA, I’, 6)
(7a)
N,(s) = C,, U,(s) + CrOV&Y),
(7b)
where DrOE R” xm denotes the highest-column-degree denotes the lowest-column-degree coefficient matrix, and C,, is a p x m constant matrix. We define the core realization as follows : Vol. 330. No 5, pp X15-839. Prinled m Great Rrllaln
coefficient matrix, D,, E R” xn Cr,, is ap x n constant matrix,
lYY3
819
W. S. Chen and J. S. H. Tsai E, = Z, (an n x rn dimensional
-0
1
. .
0
0
..
. . :.
.
.
.. A,=blockdiag
BL = blockdiag[O
K = PcDr~Qc =
@a>
matrix)
,
0
0
...
1
0
0
0
...
0
1
0
0
...
0
0 _ Kc,XKC,
,...,
O,l],,
z,
0
o
o
[
1 ,
c‘,,
i=
I,2 ,...,
i=1,2
,...,
m
(8b)
m
(EC)
w = rank [&I,
where Z,, denotes the w x w dimensional identity matrix. If the given right MFD is column-proper, then H, = I,,,, P, = D,i’, Qc = Z,. However, when the MFD is column-pseudoproper or nonstrictly-proper, then P, and Qc are arbitrarily chosen based on H, and Dfl. With the definitions of above core realization, we can obtain a generalized realization for the given system (3a) in a form of (I), where E=
E,+B,H,B:-B,Bt
(94
A = A,G, - B,P,D,,G,
(9b)
B = B,P,
(9c)
C = C,,G,+C,,Q,B:~,
(94
EE R”““, A ERGS”, BE R”““, CE Rpxn, (d(*)/dt) and G, is defined as
denotes
the time derivative
of (*),
Pe> Proof:
Now we show that C(s)(sE-AA)-‘B=
N,(s)D,‘(s).
(IO)
Of course, we firstly have to show that the matrix (SE- A) is nonsingular ; i.e. the determinant of (sE-A) is not equal to zero. Since N,(s) = C,,U,(.r)+CrOI/r(s), there exists a finite s, satisfying C(s) = C,,,G,+ C,, Q,BEs, and D,( s> 1snonsingular, det D,(s,) # 0, and D,(s,) has an inverse matrix. Substituting s, into (10) and from the core realization, we have
[G,Q,Bb,
+G,Gl(~,E-C’B
Equation
(1 la) can be divided
= [C,,~,(s,)+C,o~,(s,)lD,‘(s,).
(114
into two parts as follows,
G,Q,B:s,(s,E--A)p’B
= G ur(s,Pr-
‘@,I
(1 lb)
and Journal
820
of the Franklin lnst~tute Pergamon Press Ltd
Matrix CroG,(s,E-A)-‘B= Since U,(s’) = Q&G;‘s,
Fraction Descriptions
C,,V,(s,)D;‘(s,).
(1 Ic)
Vr(s,), (1 lb) becomes
Q,B;s,(~,E-A)~‘B
= Q,B:G,‘s,
V,(s,)D,‘(s,).
(IId)
Then, one has B = (s,E-A)G,’ Obviously, by simple is singular, then det (s,E-A) is singular, such that q(s,E-A)
V,(s,)D;
(lie)
manipulation, (1 lc) can also be reduced to (1 le). If (SE-A) (sE- A) = 0 for all finite s. Of course, for a special s,, the too. Hence, there exists a nonzero vector r of dimension 1 x n = 0. Premultiplying both sides of (1 le) by q, yields qB = r](s,E-A)G,
Combining
‘(3,).
Y](.Y’E-A) = 0 together
’ V,(s,)Dr
‘(s,) = 0.
(12)
with (12), we have
y[s,E-A
B] = 0.
(13)
This contradicts our previous assumption, thus we can deduce a conclusion from premises that (SE- A) is nonsingular. Due to the above proof, we go on to show the relationship of (10). Recalling (1 le), we thus have ’ V,(s) = BD,(s).
(sfi-A)G, Substituting
(9a) and (9b) into the left-hand
@E-&G,
V,(s) = [s(Z,+B,H,B:.-B,B;) = [s(Z, - B,B:)G,
’ -A,]
(14)
side of (14) gives
-A,G,+BJ’,D,,G,]G,’ Vr(s) + [s(B,H,B:)G,
V&s) ’ + B,f’,D,,]l/,(s). (1%
Inspecting the structures of B,, G, and A,, we can prove that the first term of (15a) is a null matrix. Hence, one has (sE- A)G, For the right-hand
’ V,(s) = [s(B,H,B:)G,
’ + BJ’,D,,]
Vr(s).
(15b)
side of (14), we have B&(J)
= BJ’,[D,,~,@)
+D,’
= &~cP,oQJW, = bW’,&Q,BW, = [s(B,H,B:)G, ’ +
VrMl
‘~f’r($+Dr’f’,(s)1 ’ +&f’c&I v,(s) BJ’,D,,]
V,(s).
(15c)
Obviously (15b) is equal to (1%). It implies that (9) is surely the generalized realization of the given right MFD in (3a). As for the controllability and observability of the realized system, according to Lemma 2.1 and in the sense of Rosenbrock and Cobb, the proofs are straightforward from the structures of the core realization matrices E,, A, and B,. It is checked that Vol. 330, No. 5, pp. 8’5-839, Printed in Great Bntam
,993
821
W. S. Chen and J. S. H. Tsai B,] = n
rank [E, rank [SE, - A,
(16)
B,] = n.
(17)
By Lemma 2.1, we know that both [E B] and [sE- A controllable. That is, they are full rank as a consequence rank[E
B] = rank [E,+B,H,Bi-
B,-
B,Bk
B] are full rank II if it is of (16) and (17),
B,P,] (18a) 1
However, P, is an elementary and nonsingular [E B] has full rank ~1.Next, one has rank [sE-A
and rank structures
B] = rank [s(E,+ B,H,Bk-
matrix,
then it is obvious
B,Bi)-AA,G,+BcP,D,,G,
= rank [SE, - A,G,
B,]
= rank [SE, - A,G,
B,..,
that
B,P,]
I
0
s(H, B: - B;) + P,D,, G,
P,
1 (18b)
[sE,-A,G, B,] is full rank for all finite s. Furthermore, from of A,, G, and B,, it implies that (SE-A) and B are left coprime.
the
Theorem I The triple {E, A, C(s)} is observable at finite modes ; i.e. the [(sE- A)’ C(s)‘]’ D,(s)] is right is full rank for all finite s if and only if polynomial matrix of [N,(s) coprime. Proof : Firstly, we show the necessary condition. If the [(sE- A)’ C(s)‘]’ pair C(s)] is a polynomial matrix pair of is full rank for all finite s, then [@E-A) right coprime. Hence, there exist polynomial matrices X,,(s) and Y,,(s), such that X,, (s)(.sEPostmultiplying
both sides of (19) by G;‘V,(s), X,, (s)(sE-
Substituting
A) + Y,, (s)C(s)
= I,.
(19)
we obtain
A)Gc- ’ V,(s) + Y,, C(s)G,
’ V,(s) = G, ’ V,(s).
(20)
(7b), (9d) and (14) into (20) yields X,, (s) BD, (s) + Y,, (s)N, (s) = G, ’ V,(s).
(21)
Checking the structures of G; ’ V,.(s) and D,(s), we know G, ’ Vr(.s) and D,(s) are right coprime. Hence, there exist polynomial matrices X,,(s) and Y+(s), such that X,,(s)G, Substituting
’ VJs)+
Y,,(s)D,(.s) = Z,.
(224
Y,,(~)I~~(s)+[X~~(S)Y,,(.~)IN,(~) = 6,.
Wb)
(21) into (22a), we get tXr2(~)J’,,(s)B+
Therefore, the D,(s) and N,(s) are right coprime. Secondly, we show the sufficient condition.
If
(N,(s)
Dr(.y)} are Journal
822
right
oSthc Franklin lnrlitute Pcrgamon Press Ltd
Fraction Descriptions
Matrix coprime, [(AE-A)’
at some finite but {E, A, C(s)} is unobservable C(>ti)‘]’< n, there exists a nonzero vector 4, such that
[
1.E- A c(n)
j,;
i.e.
rank
1
(23)
r = 0.
From (23), we can write [I-E-A
-B]
: [I
= 0,
(24)
and from (24), we have
(25) where the dimension
of [(I-E- A) -B]
is IZx (n + m) and D’(i)]
rank [(G; ’ V(i))’ From the left-hand side of (24), applying G;- ’ V,(1), we obtain rank [i.E- A
= m.
algebraic
-B]+nullity[/1E-A
(26)
theory
-B]
to the structure
= n+m.
of
(27)
-B] = PH.Furthermore, -B] = n, we have nullity [LE-A Since rank [I-E- A D:(2)] because G, is nonsingular and rank [V’/:(j_)G;’ D:(i)]’ = nz, [V:(J_)G,’ -B]. The vector [[’ 0’1’ will lie in can form all null space of [(3.E-A) -B], and it will be a linear combination of the null space of [(12-A) [W)G, ’ D:(lL)]‘. On the other hand, there exists a nonzero vector $ of dimension WIx 1, such that
(28) Premultiplying
both sides of (28) by diag {C(j_),
As C(i.) = 0 and C(I.)G,
I), yields
’ I’,(;_) = NI-(n), (29a) becomes N,.(i) D
[
(i)
T
1
$ = 0.
(29b)
D,(s)} are Clearly, (29b) is contradictive to our assumption condition that (N,(s) right coprime. Finally, we show that det (sE- A) = det D,(s). As the results of the above proofs and facts show in Lemma 2.1 and (26)
W. S. Chen und J. S. H. Tsui
rank [Gzrz;S)] we thus have the following
= m,
relationships {.sE- A
{G, ’ V,(s) That is, there exist polynomial that
rank [sE--A
B] = 12,
: (31a)
B) are left coprime
D,(s)} are right coprime.
matrices
(3tb)
{Xr3(s), Y,,(s)} and {X,,(S), Yr,(s)> such
(sE-A)X,,(s)+BY,,(s)
= I
(32a)
= I,
(32b)
where 1 is an identity matrix with an appropriate dimension. (SE-A)G, ’ Vr(.s) = BD,(s) in former proof and combining
Recalling the relation together with (32), we
X,,(S)G,
’ V,(S) + Y,,(S)&(S)
get
where R,(s) is an appropriate polynomial matrix, and both sides of (33) are square matrices. Hence, by taking determinants of both sides of (33), we have the value 1 on the right-hand side. Moreover, the second matrix of the left-hand side in (33) is a unimodular matrix. Combining together with (32a) and (14), we have
Taking
determinants
of both sides of (34), gives det (SE-A)
* K = det D,(s).
(35)
Because no scaling operation is made in the realization algorithm, has to be constant 1. Hence, the assertion is proved completely. Now, the realization for a given row-pseudoproper (row-proper) fer matrix in the left MFD will be derived in the following. Dtfinition
the value K thus
n rational
3.2
Define k,., = the ith row degree of [D,(s) U,(s) = block diag {[s’l~]}, V,(s) = block diag {[Is.. and such that
..+
N,(S)],
i = 1,2,. . . ,p
i = I, 2, . ,p ‘I},
i = 1,2,. . . ,p,
trans-
A4atri.x Fraction Descriptions
u,@>D,o + J’I(s)D,,
D,(4 =
(36)
N,(s) = U(s)B,, + V,(s)B,,,
(37)
where D,, E Rpxp denotes the highest-row-degree coefficient matrix, D,, EZ?“~” denotes the lowest-row-degree coefficient matrix, B,, is a p x m constant matrix, and B,, is an n x m constant matrix. Next, we define the core realization as follows : E, = Z, (an n x m dimensional -0
A, = block diag
identity
matrix)
0
1
0
0
1
.
.
. .
_o
0
.
i= 1,2,...,p .
.
..
1
0 1 KC, x KC,
C,, = block diag [0,.. . ,0, 11, x or,, i = 1,2,
z, H0 = Q,DroPo =
[
o
(38a)
1
0 o ,
. ,p
(38b)
(38c)
Wd)
z = rank [DJ,
where Z=denotes a z x z dimensional identity matrix. If the given left MFD is rowproper, then H,, = Z,, P,, = D, ‘, Q,, = Z,. However, when the MFD is row-pseudoproper or nonstrictly-proper, P,, and Q0 are arbitrarily chosen based on H, and D,,. Therefore, with the definitions of above core realization, we can obtain a generalized realization of the given system in the following way : E = E,fC:H,C,-C:,C,
(39a)
A = G,,A,-G,D,,P,C,
(39b)
B = G,B,,+C;Q,B,,%!
(39c)
c = P”C,,
(39d)
where EER”~“, AEZY”, BER”~“‘, CER~‘~~‘, (d(*)/dt) of (*), and G, is defined as
z
=
-p
0
the time derivative
0
L 1 I,
G,
denotes
Qo
We)
Proqf : We now show that C(sE-
A)
‘B(s) = D, ‘(s)N,(s).
(40)
Of course, we first have to show that the matrix (sE- A) is nonsingular ; i.e. the determinant of (SE-A) is not equal to zero. Since N,(s) = U,(x)B,,+ V,(s)B,,, B(s) = G,B,,+ Ci,QOB,,s, and D,(s) is nonsingular, there exists a finite s, satisfying
W. S. Clzen and J. S. H. Tsui det D,(s,) # 0, and D,(s,) has an inverse from the core realization, we have C(s,6-A)~-‘(CXQ,B,,sl Equation
+G,B,,,)
(41 a) can be divided C(s,E-A)-
matrix.
Substituting
= D~‘(sI)[U,(sI)B,,
s, into
(40), and
+ Vl(s,)Blo].
(41a)
into two parts as follows : ‘C:>Q,s,B,,
= D,-‘(s,)U,(s,)B,,
(41b)
and C’(s,E-A)-‘G,,B,, Since U,(s,) = V,(s,)G,
= D, ‘(s,)V,(s,)B,,.
(41c)
‘C~,Q~,s,, (41b) becomes
C(s,E-A)-
‘C:,QOs, = D, ‘(s,) V,(s,)G,
‘C:Q,,s,.
(41d)
Thus, we have C = D; ‘(s,)V,(S,)G,‘(S,E-A). Obviously, by simple is singular, then det (s, E-A) is singular, such that (,s,E-A)p
manipulation, (41~) can also be reduced (sE- A) = 0 for all finite s. Of course, too. Hence, there exists a nonzero vector = 0. Postmultiplying both sides of (41e) Cp = D~‘(,s,)V,(S,)G,‘(S,E-A)p
Combining
(S, E-A)p
(41e)
= 0 together
to (41e). If (sE- A) for a special s,, the p of dimension n x I by p, yields
= 0.
(42)
with (42), we have
[s,E-A
C]p = 0.
(43)
This contradicts our previous assumption, thus we can deduce a conclusion from premises that (sE- A) is nonsingular. n Due to the above proof, we go on to show the relationship of (40). Recalling (4le), one thus has V,(s)G, Substituting V,(s)G,
‘(SE-A)
= D,(s)C.
(39a) and (39b) into the left-hand
(44)
side of (44), we have
‘(SE-A)
= v,(.~)Gb
‘[s(z~+fCXH,,C,-CXC,)-G,,A,,+G,,D,,P,,C,,l
= V,(.S)LTG, ‘K-CC,>-&1
+ VICT)[.SG, ‘(Ct,H,C,)+D,,P,C,l.
(W
Inspecting the structures of C,, G, and A,, we can prove that the first term of (45a) is a null matrix. Hence one has V’,(S)G, ‘(SE-A) For the right-hand
= V,(.s)[sG, ‘(C:N,C,)+D,,P,C,].
side of (44), we have D, (s) C = [G (~1D,,, + 6 6) D, I1f’oCo
= [.sV,(s)Go ‘CbQ,D,,+ 826
V,(s)D,,]P,C,
(45b)
Matrix
=
Fraction Descriptions
v,(s)bG, ‘(CZoD,,f’,G) +D,,PoGI
= V,(S)[SG,‘(CXH,C,)+DI,P,C,I.
(45c)
Obviously (45b) is equal to (45~). It implies that (39) is surely the generalized realization of the given left MFD in (3a). As for the controllability and observability of the realized system, according to Lemma 2.1 and in the sense of Rosenbrock (15), Cobb (14) and Dai (16), the proofs are straightforward from the structures of the core realization matrices E,, A, and C,. It is checked that rank [Eb
Cb]’ = n
(46a)
rank [(SE,- A,)’
Cb]’ = n.
(46b)
By Lemma 2.1, we know that both [E’ C’]’ and [(SE-A)’ it is observable. That is, they are full rank as consequences rank [E’
C’]’ = rank [(E,+ CbH,,C,-CbCO)t I
C;Ho-C:
= rank 0
P,
However, if PO is an elementary and nonsingular [E C] has full rank n. Next, one has rank [(SE- A)’
(POCO)‘]’
I[co 1.
(47)
matrix,
that
E,
then it is obvious
C’]’
= rank[(s(E,+C~H,C,-C~C,,-G,A,+G,D,,P,C,)L
= rank [(SE, - G,A,)’ and rank structures
C’]’ are full rank n if of (46a) and (46b), i.e.
Cb]‘,
(POCO)‘]’
(48)
[(sE,-G,A,)’ CL]’ is full rank for all finite s. Furthermore, from the of A,, G, and C,, it implies that (SE-A) and C are right coprime.
Theorem II The triple {E, A, B(s)j 1scontrollable at finite modes, i.e. the [SE-A B(s)] is full rank for all finite s if and only if the polynomial matrix of [N,(S) D,(s)] is left coprime. Proof: Firstly, we show the necessary condition. If the [SE-A B(s)] pair is full rank for all finite s, then [(&--A) B(s)] is a polynomial matrix pair of left coprime. Hence, there exist polynomial matrices X,, (s) and Y,, (s), such that (SE-A)X,, Premultiplying
(s) + B(s) Y,,(s) = I,.
both sides of (4) by V,(s)G;
(49)
‘, we obtain 827
W. S. Chen and J. S. H. Tsai V,(s)G,‘(sE-A)X,,(s)+ Substituting
l’,(s)G,‘B(s)Y,,(s)
= V,(s)G,‘.
(50)
(37) and (44) into (50), yields Q(WX,($+N($Y,,($
= V($G,‘.
(51)
Checking the structures of V,(S)G; ’ and D,(s), we know that V,(.r)G; ’ and D,(s) are left coprime. Hence, there exist polynomial matrices X,,(s) and Yn(s), such that 6 (s) G, ’X,, (s) + D, (s) Y,, (s) = I,. Substituting
(524
(51) into (52a), we get
Q(s)[C4, (WD(4 + Y,,(s)1+ N(S)[Yl,(~)~‘2Wl= cl.
(52b)
Therefore, the D,(s) and N,(s) are left coprime. Secondly, we show the sufficient condition. at some Tf IN,(s) D,(s)1 are left coprime, but (E, A, B(s)} is uncontrollable finite I,, i.e. rank [(iE-A) B(A)] < n, then there exists a nonzero vector i, such that [[IE-A From
B(A)] = 0.
(53)
(53), we can write
(54) and from (44), we have
1 where the dimension
of [(/!E- A)’
side of (54), applying
rank [(,lE-
A)’
[
-c
is (n+p)
-C’]
rank [ P’,(A)Gd ’ For the right-hand G; ‘, we obtain
2E-A
1
o (55)
=>
x n and
a(41 = P.
algebraic
(564
theory to the structure
- Ct]’ + nullity [(RE- A)’
- Ctlt = n fp.
of V,(s)
(56b)
- Ctlt = p. Further-C,]’ = n, we have nullity [(I-E- A)’ Since rank [(/1E- A)’ more, because G, is nonsingular and rank [V,(A)G; ’ D,(A)] = p, [V,(A)G; ’ D,(A)] -Cl. The vector [[ 0] will lie in the null can form all null space of [(IE-A) -Cl, and it will be a linear combination of [ V,(A)G; ’ D,(A)]. space of [(LE- A) On the other hand, there exists a nonzero vector 4 of dimension 1 x p such that 4[v,(JJG;-’
Q(A)1 = K 01.
(57) Journalof
828
the Franklin Insulutc Pergamon Press Ltd
Matrix Postmultiplying
both sides of (57) by diag {B(I), $]k’,(4G,‘B(4
As B(I”) = 0 and V,(,I)G;
D,(41
Fraction Descriptions
I}, yields
= [UV)
(58)
01.
‘B(A) = N,(A), (58) becomes (P[&(4
Clearly, (59) is contradictive left coprime.
Q(41
(59)
= 0.
to our assumption
condition
that {N,(S)
D,(s)} are
n
Of course, if the given left MFD is row-proper, then there is no controllability problem at infinity. However, for the row-pseudoproper or the nonstrictly-proper MFD, we still need to check it. For the convenience of describing the controllability at infinite modes, we decompose B,,(D,,) into two submatrices Bloh(Dllh) and B,Ol(D,,,) ; i.e.
(604 where B,,,, E R(n~p)xm, Blole Rpxm, Dllh E R(n~p’xp and D,,,E Rpxp. Then we can write the result as the following proposition. Proposition 3.1 = p, the realization If rank [D10 B,,,-D,,,P,Q,B,,] infinite frequencies. Proof: From (39) and Lemma 2.2, we have rank ]E
4,
+A&1 =
=
in (39) is controllable
at
[QoDdo QoB,ol-QoDII,PoQoB,,l
(n-p)+rank
&~-D~d’,Q,&1.
(n-p)+rankPI,
(60b)
= p, then rank [E B,,+AB,,] = n; i.e. the Hence, if rank [D,, B,,,-D,,lP,Q,B,,] realization in (39) is controllable at infinite modes. Note that w-hen B,, = 0, this is the condition of row-pseudoproper MFD given by Tsai and Chen (8). Finally, we show that det (SE--A) = det D,(s). With the results of the above proofs and the facts shown in the Lemma 2.1 and (56a), gives rank [V(s)G; we thus have the following
’
rank [(sE- A)’
D(s)] = p,
relationships {SE-A
C,], = IZ,
(61a)
:
Cl are right coprime
(61b)
D,(s)} are left coprime.
(61~)
and ( V,(s)G; That is, there exist polynomial that “0,. 330. ho 5. pp X15-839. Pnn~cd in Gre;tt Britzun
’
matrices
(X,,(S),
Y,,(S)) and {X,,(S), Y,4(~)}, such
lYY3
829
W. S. Clzen and J. S. H. Tsui
X,,(S)(.SE-A)+
V,(s)G,~ ‘X,,(S)+D,(.S)Y,~(S) = D,(s)C
Recalling the relation V,(s)G; ‘(S-A) bining together with (62), we get Yn(.s)
x,X0) ’
L - V,(s)G,
D,(s)
(62a)
Y,,(.s)C = I
I[ c (SE- A)
= I.
(62b)
in the former
proof
and com-
-X,,(s) Y,,(s) ] = [1
(63)
“?I?
where R,(s) is an appropriate polynomial matrix, and both sides of (63) are square matrices ; hence by taking determinants of both sides of (3.57), we have the value 1 on the right-hand side. Moreover, the first matrix of the left-hand side in (63) is a unimodular matrix. Then combining together with (62a) and (44), we have
Taking
determinants
of both sides of (64), gives K*det (SE--A)
= detD,(s).
(65) thus the value K
Also no scaling operation is made in the realization algorithm, has to be constant 1. Hence, the assertion is proved completely.
IV. Illustrative Example
Consider
1
n
Examples
’(s) s0 =[ IL
a right-MFD
of the form
T(s) = N,(s)D,:
--s
(s+2)2
-(s+2)
0
s3+4s2+5s+2
sz
1 p’
It is desired to find the corresponding generalized state-space representation in the controller canonical form. From the given structures of N,(s) and D,(s), we know it is column-proper, since X,, = 2, K,. = 3 and ti, + ti? = deg det D,(s) = 5. By means of (7a), (7b) and the core realization in (S), we have D, CT) = Qu ui, (s) + Dr I vr 6~) -1
830
0
Matrix
Fraction Descriptions
N,(s) = C,“ffr(S)
0
1
0
-1
0
0
0
0
0
1
= [
1
0
s
0
0 01. s 1
11
Since DrOis full rank, one has -0
B,=
o-
-0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
1
0
l_
0
0
0
0
0
,
A,=
P, = D, ’ = H, = QC =
0
0
[0
1
0
0
1
0
0
0
‘I
10 By (9), the realization -1 0
0
0
0
0
1
0 0I
0
structures
0
0
0
o-
1
0
0
0
0
1
0
0
0
[
11
are 10
-0 -4
E=OOlOO,A=
-0
0
> o -1 () () 1 1 0 0 co= 1 000
G, =
0
1
1
0
0
0
-4
2
1
0
-+ 0
0
1
0
0
0
1
0
0
0
0
0
l_
0
01° 0 , -2
0
1
-5
O
-4
1
01O B=
1 -1
0
0 0
0 0
-0 The realization Vol.
330. No. 5. pp XI5
Pnnmi
in Great
Br~tan
and
C =
0
-1
0
0
0
0
0
1.1
1
after Fang and Chang
(12), is
x3’). IYY3
831
W. S. Chen and J. S. H. Tsai -1 0 E=
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
0
0
0
0
0
1
2
1
,
00100 0
0
0
1
0
0
0
0
0
1
0
0
0
0
A= -4
0
11
ro
and
Example 2 Consider a column-pseudoproper
C =
0
0
1
L0 0 0
right-MFD
0
5
-2
0
-4
0
-1
-4
1
1
(9)
From the given structures, it is seen that K,, = 3 and JC,~= 1, and K,, +K,. # deg det D,(s) = 2; i.e. D(s) is not column reduced, also it is shown to be columnpseudoproper based on the Definition 2.3. Using (7a), (7b) and the core realization in (8), we have
Since D,,, is not full rank and w = rank [Dro] = 1, H, =
832
1
1
0
P, = [ 0
1
1 o [
0 o
1
. When we choose
Matrix Fraction Descriptions
then we have 0
100
‘
11 0
0
1
:
& = [:
A
y]‘.
0I
G_=OlO 000 By (9), the realization
and
-1
structures
are
Note that the quadruple {E, A, B, C> is in the quasi-controller canonical form, since the given MFD is right coprime and rank [E’ C’]’ = 4 = n. Hence, the above realized system is observable at finite and infinite modes. The realization after Tan and Vandewalle (9) is
where E is not in the special form as shown in (3b). Example 3
Consider
a nonproper
transfer
matrix
T(s) = [-s’+2s2--s+4
By definition,
(13) -_s4+sz+6s+2]’ S3-S*_4
we have DrO=O,
c,, = [_y], V”l. 33”. No. 5. pp. YIS-x3'). lYY3 Prmted m Great Britain
DT, =[-4
0
2
-I]
G”=[:-:, ; -A]. 833
W. S. Chen and J. S. H. Tsai Since D,, = 0, hence H, = 0. For simplicity, we choose P, = Qc = 1, then G, = I4, and the corresponding state-space representation form is L%(t) = Ax(t)+Bu(t),
4’(t) = Cx(t) + DC(t),
where r E=
1
0
0
0
0
1
0
0
0
0
I
0
,
A=
:
1
1
B=
and
D =
Due to the given MFD, the right coprime and rank [E’ realized system is observable at finite and infinite modes. Example 4 Consider a left-MFD
i 1 0
-1
C’]’ = 4 = n, so the
of the form
T(s) = D, ’ (s)N,(s)
It is desired to find the corresponding generalized state-space representation in the observer canonical form. From the given structures of N,(s) and D,(s), we know it is row-proper since K,, = 3, K,? = 2 and ti,, + ti,? = deg det D,(s) = 5. By means of (36) and (37) and the core realization in (38), we have
N,(.s)= v,(~)B,o
r 1 =0001s [
s
s2
0
0
1 L-1
834
00
O Ol 1
0
0
0
11
2
-2
5
-1
4
0
0
4
0
4
Matrix
Fraction Descriptions
Since D,, is full rank, one has
L1
P,=D,‘=H,=Q”= -1 0 G,
By (39), the realization
E=
0
0
o-
1
0
0
0
0
0
1
-
OOlOO,B,,=
=
-1
0
1
0
0
1
0
0
0
0
0
1
0
0
o-
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
-
0
o-
0
0
-1
0
0
0
-1
1
-0
0
-2
0
2-
10-50 ,
A=
1
0
1
-4,O
0
0
0
0
l_
0
0
0’1 T-
10 0
0
are
0
B=
o-
10.
0
structures
0
and
0 0
-4 -4
C =
0 1
Example 5 Consider a row-pseudoproper
left MFD
s3+s
T(s) =
;]
s+2
= D, ’ (s)N, (s). From the given structures, it is seen that K,, = 3 and K,~ = 1, and K,~+q2 # deg det D,(s) = 2 ; i.e. D,(s) is not row reduced, also it is shown to be row-pseudoproper. Using (36) and (37) and the core realization in (38), we have “0,
33”. ho.
Prinlcd
5. pp. 815
,n Great
Bmain
839. lYY3
835
W. S. Chen and J. S. H. Tsai D,(.Y) = U,(S)D,” + V,(s)D,,
1
s
s2
0
=0001
1
0
()
1
10’
’
0
]!I
1
rl
01
Since D,, is not full rank and z = rank [Dlo] = 1, H,, =
PO=
c1 0
1 o
and
1
Q. = i _,
1
0
1
, ,
then we have
G, = and
By (39), the realization
structures
r 1
r 10
B=olo
0
0
10
00
01
00
00
10
I 0
0
are
~0 1
1
-1
0 0 0
1-l’ 11
and
c=
[:+;-I
I].
Note that the quadruple {E, A, B, C} is in the quasi-observer canonical form. Since the given MFD is left coprime, B,, = 0 and rank [Dlo BIoI] = 2 = p, then 836
Matrix Fraction Descriptions rank [E B] = 4 = n. Hence the above realized system is controllable infinite modes.
at finite and
Example 6 Consider
left MFD as follows :
a nonstrictly-proper
-2-l
s2+1
T(s) =
-s3_22s2-_s_2
[
,&$_s
-3s2-2s-2 =
-1
0 -3s2-2s-2
0
L =
_I
-3s2-2s-2
I
s2+1 L
-s2-
_s3-2s2_s_2
1
s3_s2_s
I
D; ’ (s)N,(s).
From the given MFD structure
&,=[-;
Since H, =
we have
;],
1
&=[-;
-;
0
o , we hence choose
corresponding
_;
_;I
-l/3
1
o [
_;
0
[
state-space
representation
1
1 , and the
o
Q0 = I2 and P, = form is y(t) = Cx(t),
I%(t) = Ax(t) +B,u(t)+B,ti(t), where -1 0
0
0
0
0
1
0
0
0
,A=
E= 0
0
0
1
0
0
0
0
0
0
-
1
-1
0
B,=
-213
0
0
O-
1
-213
0
0
0
0
0
0
0
2
0
0
1
0
2
0
0
1
3_
0
o-
‘0
-0 -
1
0
-2
0
-1
-1
-2
-1
,B,= -I
-1
0
0
0
0 1
and voi.
330, No. 5, pp. Xl5Ks39, 1993 Printed in Great Britain
837
W. S. Chen and J. S. H. Tsai 0 c=o
[
1
-113
0
0
0
0
0
0
1’
As the given MFD is left coprime and rank [& B,,I-~Dll,PoQoBll] = 2; i.e. rank [E B] = 5 = n, so the realized system is controllable at finite and infinite modes. Obviously, for the given coprime MFD, the minimal realization in state-space representation form can be obtained in this paper, even though [D,(s) N,(s)] is not row reduced. Note that the realization algorithm proposed in (8) does not work for this nonstrictly proper MFD.
v. Conclusions In this paper, we propose generalized, feasible and constructive algorithms to realize the column-pseudoproper(column-proper), row-pseudoproper(rowproper) and nonstrictly proper MFDs in the state-space representations with the quasi-controller(controller), the quasi-observer(observer) canonical forms and the system in (2), respectively. If the’ MFD is right(left) coprime, then the algorithm provides a minimal realization in state-space representation (i.e. the realized system is controllable and observable). Furthermore, if the right(left) MFD is column(row)-pseudoproper or nonstrictly-proper, then the realized system is controllable(observable) in the sense of Rosenbrock and Cobb, and we can use a simple method to check the observability(controllability) at infinity in this paper. Therefore, it appears ours is a simpler and more natural generalization than existing algorithms, thus we can regard these as a more general bridge of the conceptual gap between regular and singular systems.
References (1) D. G. Luenberger, “Dynamic equations Control, Vol. 22, pp. 312-321, 1977.
in descriptor form”, IEEE
Trans. Autom.
(2) G. C. Verghese, B. C. Leuy and T. Kailath, “A generalized state-space for singular system”, IEEE Trans. Autom. Control, Vol. 26, pp. 81 I-831, 1977. (3) T. F. S. Mohamed, “A new approach for designing a reduced-order controller of a linear singular system”, IEEE Trans. Autom. Control, Vol. 35, pp. 492495, 1990. (4) W. A. Wolovich, “The determination of state-space representations for linear multivariable systems”, IEEE Trans. Autom. Control, Vol. 9, pp. 97-106, 1973. (5) B. J. Bender and A. J. Laub, “The linear-quadratic optimal regulator for descriptor systems”, IEEE Trans. Autom. Control, Vol. 32, pp. 672-687, 1987. (6) J. Y. Lin and Z. H. Yang, “Optimal control problems for singular systems”, Znt. J. Control, Vol. 47, pp. 1915-1924, 1988. (7) Y. Y. Wang, S. J. Shi and Z. J. Zhang, “A descriptor-system approach to singular perturbation of linear regulators”, IEEE Trans. Autom. Control, Vol. 33, pp. 370373, 1988. (8) J. S. H. Tsai and W. S. Chen, “Generalized realization with a special coordinate for matrix fraction description”, Int. J. Systems Sci., Vol. 23, pp. 2197-2217, 1992. (9) S. Tan and J. Vandewalle, “Realization algorithm for determining generalized statespace representations”, Int. J. Control, Vol. 45, pp. 113771146, 1987.
838
Journal of the Franklin lnstltute Pergamon Press Ltd
Matrix
Fraction Descriptions
(10) S. Tan and J. Vandewalle, “A singular system realization for arbitrary matrix fraction descriptions”, IEEE Int. Symp. Circuit Systems, pp. 615-618, 1988. (11) T. Shiotsuki and S. Kawaji, “On a canonical form of descriptor systems”, Proc. 27th Conf. Deci. Control, pp. 2089-2090, 1988. (12) C. H. Fang and F. R. Chang, “Realization algorithm for constructing a controllable representation of a singular system with a special coordinate”, ht. J. Control, Vol. 50, pp. 1217-1226, 1989. (13) C. H. Fang and F. R. Chang, “A strongly observable and controllable realization of descriptor systems”, Control Theory Adv. Technol., Vol. 6, pp. 133-141, 1990. (14) D. Cobb, “Controllability, observability and duality in singular systems”, IEEE Trans. Autom. Control, Vol. 29, pp. 1076-1082, 1984. (15) H. H. Rosenbrock, “Structural properties of linear dynamic systems”, ht. J. Control, Vol. 20, pp. 191-202, 1974. (16) L. Y. Dai, “Control problem for linear systems with input derivatives control”, Znt. J. Systems Sci., Vol. 19, pp. 1645-1653, 1988. (17) D. Chen and F. S. Feng, “Invariants, canonical forms, and minimal realizations for singular systems”, 10th IFAC World Congress, Munchen, Vol. 9, pp. 121, 1987. (18) H. H. Rosenbrock and A. C. Pugh, “Contributions to a hierarchical theory of systems”, Int. J. Control. Vol. 19, pp. 845-867. 1974. Received : 20 September 1992 Accepted : 30 March 1993
Vol. 330. No. 5, pp. 815-839, 1993 Printed in Great Britain
839
Order Realization
Coordinate for
Matrix
with a Special
Fraction
Descriptions”
by w. s. CHEN and J.S.H.TSAI Control System Laboratory, Dqurtnzent ef Electrical Engineering, Nutionul Cheng Kung Unioersity, Tuinan 70101, Taiwan, Republic
ABSTRACT
structively
This paper presents
:
determining generalized
pseudoproper description is prored
state-space
or a column(roMl)-proper (MFD).
of the system
giren right(left)
and minimal realization representation from
or a nonstrictly-proper
The realized state-space
to be controlluhle
the> dimension
I.
a generalized
and observable
represent&ion
.form
algorithm for
a so-called rQht(lefi)
con-
column (row)matri.x Jiaction
with a special coordinute
in the sense of Rosenbrock
aficr reulizution
qf China
and Cobb;
is equul to the determinantal
degree
besides, of the
MFD in the normal sense.
Introduction Consider
an nth order linear, time-invariant Ei(t)
system characterized
= Ax(t) +Bu(t)
by (la)
v(t) = Wt),
(lb)
where x(t) E R”, u(t) E R”‘, y(t) E Rp, and E, A, B and C are real constant matrices with appropriate dimensions, and E is possibly singular. If the matrix E in system (1) is regular, then the system (1) is called a “regular system”. Otherwise, system (1) is called “singular”, or “generalized state space”, or “descriptor”. or “semistate”. Singular systems have received much attention due to extensive applications in many areas (14). However, some system models may not appear in the above form ; for example, the system realized by Ei(t)
= Ax(t) + B+(t)
+ B,u’“(t)
y(t) = Cx(t)+D”u(t)+D,u”‘(t)+
(24 ... +D,u’“‘(t)
(2b)
is not in the form of(l), where x(t) E R” is the state, u(t) E R”’ is the feasible control input, d”(t) denotes the set of i times continuous differentiable functions of u(t), y(t) E R”is the measure output. Here it is assumed that u(t) is sufficiently continuous differentiable. The main feature of such systems is that their input and output *This work was supported by the National Science Council of the Republic of China under contract NSC-8 1-0404-E-006-572. 815
W. S. Chen and J. S. H. Tsai include not only the function u(t), as for classical linear systems, but also its derivatives. In this paper, we propose a generalized, minimal and constructive realization algorithm that determines a controllable and observable generalized state-space system of the form shown in (2) from a column(row)-proper, a column(row)pseudoproper and a nonstrictly-proper rational transfer matrix T(s) in right(left) MFD T(s) = N,(s)&
‘(s)( =D,- ‘(s)N,(s)),
(3a)
respectively. If T(s) is column(row) proper, the realized matrix E is an identity matrix with an appropriate dimension. However, when the given MFD is column(row) pseudoproper or nonstrictly proper, E will be a singular matrix with a special coordinate form by the realization algorithm proposed in this paper. Bender and Laub (5), Lin and Yang (6) and Wang et al. (7) all decomposed the matrix E into a special form Ii
0
E= [ 0
0
1
(3b)
for the goal of system analysis and design, where I, denotes the,ix,j identity matrix andi = rank (E). By such a decomposition, the finite and infinite properties of a singular system, such as controllability, observability and stability of feedback control become more transparent and easier to understand. When T(s) is column(row) pseudoproper, the realized matrix Eobtained by the algorithm proposed in this paper is in the special coordinate form as in (3b), and the realized pair controller(quasi-right-type observer) {A, B) ({A, C)) forms a quasi-bottom-type canonical form, which has been precisely defined by Tsai and Chen (8). As for the nonstrictly proper MFD, the state-space representation in (2) is considered in this paper. The main purpose of this realization algorithm is to put more emphasis on preserving the structure characteristic of the given right(left) MFD at finite and infinite modes. In practical situations, when D,(.s)(D,(s)) is column(row) reduced, the MFD has no zero structures at infinity. However, the system (I) may result from physical modelling or from some kind of equivalent transformation, then the column(row) reduction is seldom met. Of course, the strict part can be separated by applying the division theorem for polynomial matrices, or one can obtain the column(row) reduction by employing unimodular matrices. If it is done the structural properties of the system (1) are completely altered at infinity. Therefore, there are very interesting practical applications that provide important ground for studying the realizable state-space representation of the given MFD. Recently, many algorithms have been proposed in this topic. Tan and Vandewalle (9, 10) proposed two algorithms that realize singular systems to be strongly controllable, but the realized systems are not guaranteed to be strongly observable or have any special form of E. The same deficiency appears in Shiotssuki and Kawaji’s work (11). Fang and Chang (12) have successfully improved upon the above disadvantages. However, the generalized state-space form for a given column(row)-
Matrix
Fraction Descriptions
proper,case obtained there is not in the controller(observer) canonical form. An alternative constructive algorithm has also been presented by Fang and Chang (13) for the realization of descriptor systems. In their work, as a nonstrictly-proper system is represented in the so-called pseudo-normalized left matrix fraction description (PNLMFD) form, the realized descriptor system is guaranteed to be strongly observable and controllable (strongly irreducible). However, the polynomial matrices N,(s) and D,(s) need to satisfy the following conditions: N,(s)] is row reduced. Note (i) D,(s) and N,(s) are left coprime and (ii) [D,(s) that T(s) = D, ‘(s)N,(s) may be a nonstrictly-proper rational matrix. Besides, for a T(s) = DIP’(s)N,(s), if the row degree of N,(s) is equal to the row degree of [D,(s) N,(s)] and N,(s) is also row reduced, then their algorithm (13) is a minimal realization algorithm. Note that the case of pseudo-normalized right MFD has not been discussed in their paper. Tsai and Chen (8) have successfully developed a minimal realization algorithm without satisfying the above condition (ii), [D,(s) N,(s)] is row reduced. However, in our previous work (8) the algorithm cannot be directly applied to a nonstrictlyproper MFD if it is not in the column(row)-pseudoproper form. In this paper, the minimal realization of T(s) = Dc’(s)N,(s) (=N,(s)D,‘(s)) can be obtained with the condition of left(right)-coprime row(column)-pseudoproper or nonstrictlyN,(s)] ([IV: D:]‘) is not row(column) proper rational matrix only, even if [D,(s) reduced. Moreover, if T(S) is column(row)-proper, then the generalized state-space form {A, B}({A, C}) obtained in this paper is in the standard bottom-type(righttype) controller(observer) canonical form, and the matrix E is an identity matrix with an appropriate dimension. Compared to the existing ones, ours are much more constructive and without loss of generality. Also, the realized system with a special coordinate form is guaranteed to be strongly controllable and observable if the MFD is right(left) coprime in column(row)-pseudoproper or column(row)-proper or nonstrictlyproper form. According to the definition of Cobb (14) and in the sense of Rosenbrock (15) and Dai (16), the controllability and observability conditions can be easily derived by using these structures and they become more easily checked by the core realization proposed in this paper. II. Preliminaries Firstly,
let us review some properties
of the MFD.
Dqfinition 2.1
If a square polynomial matrix R(s) is the common right(left) divisor and it is also the left(right) multipler of each common right(left) divisor of N(s) and D(s), then R(s) is the greatest common right(left) divisor of N(s) and D(s). If R(s) is a unimodular matrix, then N(s) and D(s) are right(left) coprime. DeJinition 2.2 (8)
Consider
a nonsingular
polynomial
matrix D(s) with a dimension
K,, = the highest degree of the ith column
of D(s)
p x p, and let
W. S. Chen and J. S. H. Tsai 4,
= the highest degree of the ith row of D(s).
If degdet then D(s) is column
reduced.
D(s) = c ICY), *=I
If degdet
D(s) = f; tirir i= I
then D(s) is row reduced. D
of the form ’+
+d(,,,,
(4)
where each coefficient d,, (j = 0, 1, , I?) belongs to a real field. Here d,, may not be required to be nonzero and q will be called the generalized degree of the polynomial in (4) in order to distinguish it from the normal definition of polynomial degree. To construct a p x II? polynomial matrix in a so-called column-based form, they choose p.~ polynomials from Fk, [s] and arrange them into the first column such that [d, (s)dz(s) .
d,,(s)]’ = [doIdol. . . do,,]‘.+ + [d, ,d,,.
d,,,]‘.rkl ’
+‘..+[d,,,d~,12...d,,,,1’,
(5a)
where the superscript “t” denotes transpose. As in (5a), one again chooses ps polynomials from FA1[s] and arranges them into the second column, carrying out these procedures until nz columns are entirely constructed. The polynomial matrix constructed in this way is said to be in column-based form. The indices k,, which may be all different, denote the ith generalized column degree of the polynomial matrix in column-based form. The matrix polynomial, already used widely in the literature (17, 18) whose forms are normally described as D(s) = D,,+D,.s+Dzs’+
... +D,,f,
(5b)
where D(s) E R[s]“““‘, D, E RPX”’ may be viewed as a special case of the polynomial matrix in column-based form where the generalized column degrees are all the same. They also call the degree of the determinant of a polynomial matrix, in column-based form, the generalized determinantal degree. It is equal to the sum of all generalized column degrees. Assume the given right MFD (3a) is already in column-based form, then it will be called column pseudoproper if k, > I’, (i = 1, 2, . . . . m) where k, and Y, are the ith generalized column degrees of D(s) and N(s), respectively. Similarly, these definitions can be appropriately extended to those of the row-pseudoproper cases.
Matrix
Fraction Descriptions
Definition 2.4 (8) In Definition 2.3, if the term D(s) is column reduced, then the given right MFD (3a) is called a column-proper right MFD. Similarly, this definition can be appropriately extended to the row-proper case. Following Tan and Vandewalle (9, lo), we assume the minimal value of ki is zero in this paper. Note that polynomials d,(s) = 0 - +s+ 1 E F2[s] and d2(.s) = s+ 1 E F, [s] from two different sets are not the same. Naturally, a nonzero scalar dgF,,[s] should not be confused with the polynomial d,(s) = O~syO*sy -‘f ... + 0 - s + d which is an element of F,[s]. These definitions ensure algebraically that the operations involved do not destroy system behaviour either at finite or infinite frequencies.
s*
Lemma 2.1 (14, 15) Consider the system (1). (i) It is controllable for all s. (ii) It is observable rank for all s.
B] and [E
if and only if both [&-A if and only if both
B] are full rank
C’]’ and [E’
[(X-A)’
Ctlt are full
Lemma 2.2 (16) The system i(t)
= Ax(t)+B,u(t)+B,ti(t)
(64
(6b)
y(t) = C-x(t) is state controllable if and only if %![B,,+AB,, x(t) E R” and B?(o) represents the range space of
A(B,+AB,), l
B,] = R”, where
.
III. Main Results Dqfinition 3.1 Define k,, = the ith column
degree of [Or(s)
Ur(s) = block diag([,+]},i= V:(s) = block diag ([ls..
N,(s)], i = 1,2,
. ,m
1,2,...,m .,+-‘]},i
= 1,2,. . . ,m,
and such that D, (7) = L&Ou,(s) + LA, I’, 6)
(7a)
N,(s) = C,, U,(s) + CrOV&Y),
(7b)
where DrOE R” xm denotes the highest-column-degree denotes the lowest-column-degree coefficient matrix, and C,, is a p x m constant matrix. We define the core realization as follows : Vol. 330. No 5, pp X15-839. Prinled m Great Rrllaln
coefficient matrix, D,, E R” xn Cr,, is ap x n constant matrix,
lYY3
819
W. S. Chen and J. S. H. Tsai E, = Z, (an n x rn dimensional
-0
1
. .
0
0
..
. . :.
.
.
.. A,=blockdiag
BL = blockdiag[O
K = PcDr~Qc =
@a>
matrix)
,
0
0
...
1
0
0
0
...
0
1
0
0
...
0
0 _ Kc,XKC,
,...,
O,l],,
z,
0
o
o
[
1 ,
c‘,,
i=
I,2 ,...,
i=1,2
,...,
m
(8b)
m
(EC)
w = rank [&I,
where Z,, denotes the w x w dimensional identity matrix. If the given right MFD is column-proper, then H, = I,,,, P, = D,i’, Qc = Z,. However, when the MFD is column-pseudoproper or nonstrictly-proper, then P, and Qc are arbitrarily chosen based on H, and Dfl. With the definitions of above core realization, we can obtain a generalized realization for the given system (3a) in a form of (I), where E=
E,+B,H,B:-B,Bt
(94
A = A,G, - B,P,D,,G,
(9b)
B = B,P,
(9c)
C = C,,G,+C,,Q,B:~,
(94
EE R”““, A ERGS”, BE R”““, CE Rpxn, (d(*)/dt) and G, is defined as
denotes
the time derivative
of (*),
Pe> Proof:
Now we show that C(s)(sE-AA)-‘B=
N,(s)D,‘(s).
(IO)
Of course, we firstly have to show that the matrix (SE- A) is nonsingular ; i.e. the determinant of (sE-A) is not equal to zero. Since N,(s) = C,,U,(.r)+CrOI/r(s), there exists a finite s, satisfying C(s) = C,,,G,+ C,, Q,BEs, and D,( s> 1snonsingular, det D,(s,) # 0, and D,(s,) has an inverse matrix. Substituting s, into (10) and from the core realization, we have
[G,Q,Bb,
+G,Gl(~,E-C’B
Equation
(1 la) can be divided
= [C,,~,(s,)+C,o~,(s,)lD,‘(s,).
(114
into two parts as follows,
G,Q,B:s,(s,E--A)p’B
= G ur(s,Pr-
‘@,I
(1 lb)
and Journal
820
of the Franklin lnst~tute Pergamon Press Ltd
Matrix CroG,(s,E-A)-‘B= Since U,(s’) = Q&G;‘s,
Fraction Descriptions
C,,V,(s,)D;‘(s,).
(1 Ic)
Vr(s,), (1 lb) becomes
Q,B;s,(~,E-A)~‘B
= Q,B:G,‘s,
V,(s,)D,‘(s,).
(IId)
Then, one has B = (s,E-A)G,’ Obviously, by simple is singular, then det (s,E-A) is singular, such that q(s,E-A)
V,(s,)D;
(lie)
manipulation, (1 lc) can also be reduced to (1 le). If (SE-A) (sE- A) = 0 for all finite s. Of course, for a special s,, the too. Hence, there exists a nonzero vector r of dimension 1 x n = 0. Premultiplying both sides of (1 le) by q, yields qB = r](s,E-A)G,
Combining
‘(3,).
Y](.Y’E-A) = 0 together
’ V,(s,)Dr
‘(s,) = 0.
(12)
with (12), we have
y[s,E-A
B] = 0.
(13)
This contradicts our previous assumption, thus we can deduce a conclusion from premises that (SE- A) is nonsingular. Due to the above proof, we go on to show the relationship of (10). Recalling (1 le), we thus have ’ V,(s) = BD,(s).
(sfi-A)G, Substituting
(9a) and (9b) into the left-hand
@E-&G,
V,(s) = [s(Z,+B,H,B:.-B,B;) = [s(Z, - B,B:)G,
’ -A,]
(14)
side of (14) gives
-A,G,+BJ’,D,,G,]G,’ Vr(s) + [s(B,H,B:)G,
V&s) ’ + B,f’,D,,]l/,(s). (1%
Inspecting the structures of B,, G, and A,, we can prove that the first term of (15a) is a null matrix. Hence, one has (sE- A)G, For the right-hand
’ V,(s) = [s(B,H,B:)G,
’ + BJ’,D,,]
Vr(s).
(15b)
side of (14), we have B&(J)
= BJ’,[D,,~,@)
+D,’
= &~cP,oQJW, = bW’,&Q,BW, = [s(B,H,B:)G, ’ +
VrMl
‘~f’r($+Dr’f’,(s)1 ’ +&f’c&I v,(s) BJ’,D,,]
V,(s).
(15c)
Obviously (15b) is equal to (1%). It implies that (9) is surely the generalized realization of the given right MFD in (3a). As for the controllability and observability of the realized system, according to Lemma 2.1 and in the sense of Rosenbrock and Cobb, the proofs are straightforward from the structures of the core realization matrices E,, A, and B,. It is checked that Vol. 330, No. 5, pp. 8’5-839, Printed in Great Bntam
,993
821
W. S. Chen and J. S. H. Tsai B,] = n
rank [E, rank [SE, - A,
(16)
B,] = n.
(17)
By Lemma 2.1, we know that both [E B] and [sE- A controllable. That is, they are full rank as a consequence rank[E
B] = rank [E,+B,H,Bi-
B,-
B,Bk
B] are full rank II if it is of (16) and (17),
B,P,] (18a) 1
However, P, is an elementary and nonsingular [E B] has full rank ~1.Next, one has rank [sE-A
and rank structures
B] = rank [s(E,+ B,H,Bk-
matrix,
then it is obvious
B,Bi)-AA,G,+BcP,D,,G,
= rank [SE, - A,G,
B,]
= rank [SE, - A,G,
B,..,
that
B,P,]
I
0
s(H, B: - B;) + P,D,, G,
P,
1 (18b)
[sE,-A,G, B,] is full rank for all finite s. Furthermore, from of A,, G, and B,, it implies that (SE-A) and B are left coprime.
the
Theorem I The triple {E, A, C(s)} is observable at finite modes ; i.e. the [(sE- A)’ C(s)‘]’ D,(s)] is right is full rank for all finite s if and only if polynomial matrix of [N,(s) coprime. Proof : Firstly, we show the necessary condition. If the [(sE- A)’ C(s)‘]’ pair C(s)] is a polynomial matrix pair of is full rank for all finite s, then [@E-A) right coprime. Hence, there exist polynomial matrices X,,(s) and Y,,(s), such that X,, (s)(.sEPostmultiplying
both sides of (19) by G;‘V,(s), X,, (s)(sE-
Substituting
A) + Y,, (s)C(s)
= I,.
(19)
we obtain
A)Gc- ’ V,(s) + Y,, C(s)G,
’ V,(s) = G, ’ V,(s).
(20)
(7b), (9d) and (14) into (20) yields X,, (s) BD, (s) + Y,, (s)N, (s) = G, ’ V,(s).
(21)
Checking the structures of G; ’ V,.(s) and D,(s), we know G, ’ Vr(.s) and D,(s) are right coprime. Hence, there exist polynomial matrices X,,(s) and Y+(s), such that X,,(s)G, Substituting
’ VJs)+
Y,,(s)D,(.s) = Z,.
(224
Y,,(~)I~~(s)+[X~~(S)Y,,(.~)IN,(~) = 6,.
Wb)
(21) into (22a), we get tXr2(~)J’,,(s)B+
Therefore, the D,(s) and N,(s) are right coprime. Secondly, we show the sufficient condition.
If
(N,(s)
Dr(.y)} are Journal
822
right
oSthc Franklin lnrlitute Pcrgamon Press Ltd
Fraction Descriptions
Matrix coprime, [(AE-A)’
at some finite but {E, A, C(s)} is unobservable C(>ti)‘]’< n, there exists a nonzero vector 4, such that
[
1.E- A c(n)
j,;
i.e.
rank
1
(23)
r = 0.
From (23), we can write [I-E-A
-B]
: [I
= 0,
(24)
and from (24), we have
(25) where the dimension
of [(I-E- A) -B]
is IZx (n + m) and D’(i)]
rank [(G; ’ V(i))’ From the left-hand side of (24), applying G;- ’ V,(1), we obtain rank [i.E- A
= m.
algebraic
-B]+nullity[/1E-A
(26)
theory
-B]
to the structure
= n+m.
of
(27)
-B] = PH.Furthermore, -B] = n, we have nullity [LE-A Since rank [I-E- A D:(2)] because G, is nonsingular and rank [V’/:(j_)G;’ D:(i)]’ = nz, [V:(J_)G,’ -B]. The vector [[’ 0’1’ will lie in can form all null space of [(3.E-A) -B], and it will be a linear combination of the null space of [(12-A) [W)G, ’ D:(lL)]‘. On the other hand, there exists a nonzero vector $ of dimension WIx 1, such that
(28) Premultiplying
both sides of (28) by diag {C(j_),
As C(i.) = 0 and C(I.)G,
I), yields
’ I’,(;_) = NI-(n), (29a) becomes N,.(i) D
[
(i)
T
1
$ = 0.
(29b)
D,(s)} are Clearly, (29b) is contradictive to our assumption condition that (N,(s) right coprime. Finally, we show that det (sE- A) = det D,(s). As the results of the above proofs and facts show in Lemma 2.1 and (26)
W. S. Chen und J. S. H. Tsui
rank [Gzrz;S)] we thus have the following
= m,
relationships {.sE- A
{G, ’ V,(s) That is, there exist polynomial that
rank [sE--A
B] = 12,
: (31a)
B) are left coprime
D,(s)} are right coprime.
matrices
(3tb)
{Xr3(s), Y,,(s)} and {X,,(S), Yr,(s)> such
(sE-A)X,,(s)+BY,,(s)
= I
(32a)
= I,
(32b)
where 1 is an identity matrix with an appropriate dimension. (SE-A)G, ’ Vr(.s) = BD,(s) in former proof and combining
Recalling the relation together with (32), we
X,,(S)G,
’ V,(S) + Y,,(S)&(S)
get
where R,(s) is an appropriate polynomial matrix, and both sides of (33) are square matrices. Hence, by taking determinants of both sides of (33), we have the value 1 on the right-hand side. Moreover, the second matrix of the left-hand side in (33) is a unimodular matrix. Combining together with (32a) and (14), we have
Taking
determinants
of both sides of (34), gives det (SE-A)
* K = det D,(s).
(35)
Because no scaling operation is made in the realization algorithm, has to be constant 1. Hence, the assertion is proved completely. Now, the realization for a given row-pseudoproper (row-proper) fer matrix in the left MFD will be derived in the following. Dtfinition
the value K thus
n rational
3.2
Define k,., = the ith row degree of [D,(s) U,(s) = block diag {[s’l~]}, V,(s) = block diag {[Is.. and such that
..+
N,(S)],
i = 1,2,. . . ,p
i = I, 2, . ,p ‘I},
i = 1,2,. . . ,p,
trans-
A4atri.x Fraction Descriptions
u,@>D,o + J’I(s)D,,
D,(4 =
(36)
N,(s) = U(s)B,, + V,(s)B,,,
(37)
where D,, E Rpxp denotes the highest-row-degree coefficient matrix, D,, EZ?“~” denotes the lowest-row-degree coefficient matrix, B,, is a p x m constant matrix, and B,, is an n x m constant matrix. Next, we define the core realization as follows : E, = Z, (an n x m dimensional -0
A, = block diag
identity
matrix)
0
1
0
0
1
.
.
. .
_o
0
.
i= 1,2,...,p .
.
..
1
0 1 KC, x KC,
C,, = block diag [0,.. . ,0, 11, x or,, i = 1,2,
z, H0 = Q,DroPo =
[
o
(38a)
1
0 o ,
. ,p
(38b)
(38c)
Wd)
z = rank [DJ,
where Z=denotes a z x z dimensional identity matrix. If the given left MFD is rowproper, then H,, = Z,, P,, = D, ‘, Q,, = Z,. However, when the MFD is row-pseudoproper or nonstrictly-proper, P,, and Q0 are arbitrarily chosen based on H, and D,,. Therefore, with the definitions of above core realization, we can obtain a generalized realization of the given system in the following way : E = E,fC:H,C,-C:,C,
(39a)
A = G,,A,-G,D,,P,C,
(39b)
B = G,B,,+C;Q,B,,%!
(39c)
c = P”C,,
(39d)
where EER”~“, AEZY”, BER”~“‘, CER~‘~~‘, (d(*)/dt) of (*), and G, is defined as
z
=
-p
0
the time derivative
0
L 1 I,
G,
denotes
Qo
We)
Proqf : We now show that C(sE-
A)
‘B(s) = D, ‘(s)N,(s).
(40)
Of course, we first have to show that the matrix (sE- A) is nonsingular ; i.e. the determinant of (SE-A) is not equal to zero. Since N,(s) = U,(x)B,,+ V,(s)B,,, B(s) = G,B,,+ Ci,QOB,,s, and D,(s) is nonsingular, there exists a finite s, satisfying
W. S. Clzen and J. S. H. Tsui det D,(s,) # 0, and D,(s,) has an inverse from the core realization, we have C(s,6-A)~-‘(CXQ,B,,sl Equation
+G,B,,,)
(41 a) can be divided C(s,E-A)-
matrix.
Substituting
= D~‘(sI)[U,(sI)B,,
s, into
(40), and
+ Vl(s,)Blo].
(41a)
into two parts as follows : ‘C:>Q,s,B,,
= D,-‘(s,)U,(s,)B,,
(41b)
and C’(s,E-A)-‘G,,B,, Since U,(s,) = V,(s,)G,
= D, ‘(s,)V,(s,)B,,.
(41c)
‘C~,Q~,s,, (41b) becomes
C(s,E-A)-
‘C:,QOs, = D, ‘(s,) V,(s,)G,
‘C:Q,,s,.
(41d)
Thus, we have C = D; ‘(s,)V,(S,)G,‘(S,E-A). Obviously, by simple is singular, then det (s, E-A) is singular, such that (,s,E-A)p
manipulation, (41~) can also be reduced (sE- A) = 0 for all finite s. Of course, too. Hence, there exists a nonzero vector = 0. Postmultiplying both sides of (41e) Cp = D~‘(,s,)V,(S,)G,‘(S,E-A)p
Combining
(S, E-A)p
(41e)
= 0 together
to (41e). If (sE- A) for a special s,, the p of dimension n x I by p, yields
= 0.
(42)
with (42), we have
[s,E-A
C]p = 0.
(43)
This contradicts our previous assumption, thus we can deduce a conclusion from premises that (sE- A) is nonsingular. n Due to the above proof, we go on to show the relationship of (40). Recalling (4le), one thus has V,(s)G, Substituting V,(s)G,
‘(SE-A)
= D,(s)C.
(39a) and (39b) into the left-hand
(44)
side of (44), we have
‘(SE-A)
= v,(.~)Gb
‘[s(z~+fCXH,,C,-CXC,)-G,,A,,+G,,D,,P,,C,,l
= V,(.S)LTG, ‘K-CC,>-&1
+ VICT)[.SG, ‘(Ct,H,C,)+D,,P,C,l.
(W
Inspecting the structures of C,, G, and A,, we can prove that the first term of (45a) is a null matrix. Hence one has V’,(S)G, ‘(SE-A) For the right-hand
= V,(.s)[sG, ‘(C:N,C,)+D,,P,C,].
side of (44), we have D, (s) C = [G (~1D,,, + 6 6) D, I1f’oCo
= [.sV,(s)Go ‘CbQ,D,,+ 826
V,(s)D,,]P,C,
(45b)
Matrix
=
Fraction Descriptions
v,(s)bG, ‘(CZoD,,f’,G) +D,,PoGI
= V,(S)[SG,‘(CXH,C,)+DI,P,C,I.
(45c)
Obviously (45b) is equal to (45~). It implies that (39) is surely the generalized realization of the given left MFD in (3a). As for the controllability and observability of the realized system, according to Lemma 2.1 and in the sense of Rosenbrock (15), Cobb (14) and Dai (16), the proofs are straightforward from the structures of the core realization matrices E,, A, and C,. It is checked that rank [Eb
Cb]’ = n
(46a)
rank [(SE,- A,)’
Cb]’ = n.
(46b)
By Lemma 2.1, we know that both [E’ C’]’ and [(SE-A)’ it is observable. That is, they are full rank as consequences rank [E’
C’]’ = rank [(E,+ CbH,,C,-CbCO)t I
C;Ho-C:
= rank 0
P,
However, if PO is an elementary and nonsingular [E C] has full rank n. Next, one has rank [(SE- A)’
(POCO)‘]’
I[co 1.
(47)
matrix,
that
E,
then it is obvious
C’]’
= rank[(s(E,+C~H,C,-C~C,,-G,A,+G,D,,P,C,)L
= rank [(SE, - G,A,)’ and rank structures
C’]’ are full rank n if of (46a) and (46b), i.e.
Cb]‘,
(POCO)‘]’
(48)
[(sE,-G,A,)’ CL]’ is full rank for all finite s. Furthermore, from the of A,, G, and C,, it implies that (SE-A) and C are right coprime.
Theorem II The triple {E, A, B(s)j 1scontrollable at finite modes, i.e. the [SE-A B(s)] is full rank for all finite s if and only if the polynomial matrix of [N,(S) D,(s)] is left coprime. Proof: Firstly, we show the necessary condition. If the [SE-A B(s)] pair is full rank for all finite s, then [(&--A) B(s)] is a polynomial matrix pair of left coprime. Hence, there exist polynomial matrices X,, (s) and Y,, (s), such that (SE-A)X,, Premultiplying
(s) + B(s) Y,,(s) = I,.
both sides of (4) by V,(s)G;
(49)
‘, we obtain 827
W. S. Chen and J. S. H. Tsai V,(s)G,‘(sE-A)X,,(s)+ Substituting
l’,(s)G,‘B(s)Y,,(s)
= V,(s)G,‘.
(50)
(37) and (44) into (50), yields Q(WX,($+N($Y,,($
= V($G,‘.
(51)
Checking the structures of V,(S)G; ’ and D,(s), we know that V,(.r)G; ’ and D,(s) are left coprime. Hence, there exist polynomial matrices X,,(s) and Yn(s), such that 6 (s) G, ’X,, (s) + D, (s) Y,, (s) = I,. Substituting
(524
(51) into (52a), we get
Q(s)[C4, (WD(4 + Y,,(s)1+ N(S)[Yl,(~)~‘2Wl= cl.
(52b)
Therefore, the D,(s) and N,(s) are left coprime. Secondly, we show the sufficient condition. at some Tf IN,(s) D,(s)1 are left coprime, but (E, A, B(s)} is uncontrollable finite I,, i.e. rank [(iE-A) B(A)] < n, then there exists a nonzero vector i, such that [[IE-A From
B(A)] = 0.
(53)
(53), we can write
(54) and from (44), we have
1 where the dimension
of [(/!E- A)’
side of (54), applying
rank [(,lE-
A)’
[
-c
is (n+p)
-C’]
rank [ P’,(A)Gd ’ For the right-hand G; ‘, we obtain
2E-A
1
o (55)
=>
x n and
a(41 = P.
algebraic
(564
theory to the structure
- Ct]’ + nullity [(RE- A)’
- Ctlt = n fp.
of V,(s)
(56b)
- Ctlt = p. Further-C,]’ = n, we have nullity [(I-E- A)’ Since rank [(/1E- A)’ more, because G, is nonsingular and rank [V,(A)G; ’ D,(A)] = p, [V,(A)G; ’ D,(A)] -Cl. The vector [[ 0] will lie in the null can form all null space of [(IE-A) -Cl, and it will be a linear combination of [ V,(A)G; ’ D,(A)]. space of [(LE- A) On the other hand, there exists a nonzero vector 4 of dimension 1 x p such that 4[v,(JJG;-’
Q(A)1 = K 01.
(57) Journalof
828
the Franklin Insulutc Pergamon Press Ltd
Matrix Postmultiplying
both sides of (57) by diag {B(I), $]k’,(4G,‘B(4
As B(I”) = 0 and V,(,I)G;
D,(41
Fraction Descriptions
I}, yields
= [UV)
(58)
01.
‘B(A) = N,(A), (58) becomes (P[&(4
Clearly, (59) is contradictive left coprime.
Q(41
(59)
= 0.
to our assumption
condition
that {N,(S)
D,(s)} are
n
Of course, if the given left MFD is row-proper, then there is no controllability problem at infinity. However, for the row-pseudoproper or the nonstrictly-proper MFD, we still need to check it. For the convenience of describing the controllability at infinite modes, we decompose B,,(D,,) into two submatrices Bloh(Dllh) and B,Ol(D,,,) ; i.e.
(604 where B,,,, E R(n~p)xm, Blole Rpxm, Dllh E R(n~p’xp and D,,,E Rpxp. Then we can write the result as the following proposition. Proposition 3.1 = p, the realization If rank [D10 B,,,-D,,,P,Q,B,,] infinite frequencies. Proof: From (39) and Lemma 2.2, we have rank ]E
4,
+A&1 =
=
in (39) is controllable
at
[QoDdo QoB,ol-QoDII,PoQoB,,l
(n-p)+rank
&~-D~d’,Q,&1.
(n-p)+rankPI,
(60b)
= p, then rank [E B,,+AB,,] = n; i.e. the Hence, if rank [D,, B,,,-D,,lP,Q,B,,] realization in (39) is controllable at infinite modes. Note that w-hen B,, = 0, this is the condition of row-pseudoproper MFD given by Tsai and Chen (8). Finally, we show that det (SE--A) = det D,(s). With the results of the above proofs and the facts shown in the Lemma 2.1 and (56a), gives rank [V(s)G; we thus have the following
’
rank [(sE- A)’
D(s)] = p,
relationships {SE-A
C,], = IZ,
(61a)
:
Cl are right coprime
(61b)
D,(s)} are left coprime.
(61~)
and ( V,(s)G; That is, there exist polynomial that “0,. 330. ho 5. pp X15-839. Pnn~cd in Gre;tt Britzun
’
matrices
(X,,(S),
Y,,(S)) and {X,,(S), Y,4(~)}, such
lYY3
829
W. S. Clzen and J. S. H. Tsui
X,,(S)(.SE-A)+
V,(s)G,~ ‘X,,(S)+D,(.S)Y,~(S) = D,(s)C
Recalling the relation V,(s)G; ‘(S-A) bining together with (62), we get Yn(.s)
x,X0) ’
L - V,(s)G,
D,(s)
(62a)
Y,,(.s)C = I
I[ c (SE- A)
= I.
(62b)
in the former
proof
and com-
-X,,(s) Y,,(s) ] = [1
(63)
“?I?
where R,(s) is an appropriate polynomial matrix, and both sides of (63) are square matrices ; hence by taking determinants of both sides of (3.57), we have the value 1 on the right-hand side. Moreover, the first matrix of the left-hand side in (63) is a unimodular matrix. Then combining together with (62a) and (44), we have
Taking
determinants
of both sides of (64), gives K*det (SE--A)
= detD,(s).
(65) thus the value K
Also no scaling operation is made in the realization algorithm, has to be constant 1. Hence, the assertion is proved completely.
IV. Illustrative Example
Consider
1
n
Examples
’(s) s0 =[ IL
a right-MFD
of the form
T(s) = N,(s)D,:
--s
(s+2)2
-(s+2)
0
s3+4s2+5s+2
sz
1 p’
It is desired to find the corresponding generalized state-space representation in the controller canonical form. From the given structures of N,(s) and D,(s), we know it is column-proper, since X,, = 2, K,. = 3 and ti, + ti? = deg det D,(s) = 5. By means of (7a), (7b) and the core realization in (S), we have D, CT) = Qu ui, (s) + Dr I vr 6~) -1
830
0
Matrix
Fraction Descriptions
N,(s) = C,“ffr(S)
0
1
0
-1
0
0
0
0
0
1
= [
1
0
s
0
0 01. s 1
11
Since DrOis full rank, one has -0
B,=
o-
-0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
0
0
1
0
l_
0
0
0
0
0
,
A,=
P, = D, ’ = H, = QC =
0
0
[0
1
0
0
1
0
0
0
‘I
10 By (9), the realization -1 0
0
0
0
0
1
0 0I
0
structures
0
0
0
o-
1
0
0
0
0
1
0
0
0
[
11
are 10
-0 -4
E=OOlOO,A=
-0
0
> o -1 () () 1 1 0 0 co= 1 000
G, =
0
1
1
0
0
0
-4
2
1
0
-+ 0
0
1
0
0
0
1
0
0
0
0
0
l_
0
01° 0 , -2
0
1
-5
O
-4
1
01O B=
1 -1
0
0 0
0 0
-0 The realization Vol.
330. No. 5. pp XI5
Pnnmi
in Great
Br~tan
and
C =
0
-1
0
0
0
0
0
1.1
1
after Fang and Chang
(12), is
x3’). IYY3
831
W. S. Chen and J. S. H. Tsai -1 0 E=
0
0
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
0
0
0
0
0
1
2
1
,
00100 0
0
0
1
0
0
0
0
0
1
0
0
0
0
A= -4
0
11
ro
and
Example 2 Consider a column-pseudoproper
C =
0
0
1
L0 0 0
right-MFD
0
5
-2
0
-4
0
-1
-4
1
1
(9)
From the given structures, it is seen that K,, = 3 and JC,~= 1, and K,, +K,. # deg det D,(s) = 2; i.e. D(s) is not column reduced, also it is shown to be columnpseudoproper based on the Definition 2.3. Using (7a), (7b) and the core realization in (8), we have
Since D,,, is not full rank and w = rank [Dro] = 1, H, =
832
1
1
0
P, = [ 0
1
1 o [
0 o
1
. When we choose
Matrix Fraction Descriptions
then we have 0
100
‘
11 0
0
1
:
& = [:
A
y]‘.
0I
G_=OlO 000 By (9), the realization
and
-1
structures
are
Note that the quadruple {E, A, B, C> is in the quasi-controller canonical form, since the given MFD is right coprime and rank [E’ C’]’ = 4 = n. Hence, the above realized system is observable at finite and infinite modes. The realization after Tan and Vandewalle (9) is
where E is not in the special form as shown in (3b). Example 3
Consider
a nonproper
transfer
matrix
T(s) = [-s’+2s2--s+4
By definition,
(13) -_s4+sz+6s+2]’ S3-S*_4
we have DrO=O,
c,, = [_y], V”l. 33”. No. 5. pp. YIS-x3'). lYY3 Prmted m Great Britain
DT, =[-4
0
2
-I]
G”=[:-:, ; -A]. 833
W. S. Chen and J. S. H. Tsai Since D,, = 0, hence H, = 0. For simplicity, we choose P, = Qc = 1, then G, = I4, and the corresponding state-space representation form is L%(t) = Ax(t)+Bu(t),
4’(t) = Cx(t) + DC(t),
where r E=
1
0
0
0
0
1
0
0
0
0
I
0
,
A=
:
1
1
B=
and
D =
Due to the given MFD, the right coprime and rank [E’ realized system is observable at finite and infinite modes. Example 4 Consider a left-MFD
i 1 0
-1
C’]’ = 4 = n, so the
of the form
T(s) = D, ’ (s)N,(s)
It is desired to find the corresponding generalized state-space representation in the observer canonical form. From the given structures of N,(s) and D,(s), we know it is row-proper since K,, = 3, K,? = 2 and ti,, + ti,? = deg det D,(s) = 5. By means of (36) and (37) and the core realization in (38), we have
N,(.s)= v,(~)B,o
r 1 =0001s [
s
s2
0
0
1 L-1
834
00
O Ol 1
0
0
0
11
2
-2
5
-1
4
0
0
4
0
4
Matrix
Fraction Descriptions
Since D,, is full rank, one has
L1
P,=D,‘=H,=Q”= -1 0 G,
By (39), the realization
E=
0
0
o-
1
0
0
0
0
0
1
-
OOlOO,B,,=
=
-1
0
1
0
0
1
0
0
0
0
0
1
0
0
o-
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
-
0
o-
0
0
-1
0
0
0
-1
1
-0
0
-2
0
2-
10-50 ,
A=
1
0
1
-4,O
0
0
0
0
l_
0
0
0’1 T-
10 0
0
are
0
B=
o-
10.
0
structures
0
and
0 0
-4 -4
C =
0 1
Example 5 Consider a row-pseudoproper
left MFD
s3+s
T(s) =
;]
s+2
= D, ’ (s)N, (s). From the given structures, it is seen that K,, = 3 and K,~ = 1, and K,~+q2 # deg det D,(s) = 2 ; i.e. D,(s) is not row reduced, also it is shown to be row-pseudoproper. Using (36) and (37) and the core realization in (38), we have “0,
33”. ho.
Prinlcd
5. pp. 815
,n Great
Bmain
839. lYY3
835
W. S. Chen and J. S. H. Tsai D,(.Y) = U,(S)D,” + V,(s)D,,
1
s
s2
0
=0001
1
0
()
1
10’
’
0
]!I
1
rl
01
Since D,, is not full rank and z = rank [Dlo] = 1, H,, =
PO=
c1 0
1 o
and
1
Q. = i _,
1
0
1
, ,
then we have
G, = and
By (39), the realization
structures
r 1
r 10
B=olo
0
0
10
00
01
00
00
10
I 0
0
are
~0 1
1
-1
0 0 0
1-l’ 11
and
c=
[:+;-I
I].
Note that the quadruple {E, A, B, C} is in the quasi-observer canonical form. Since the given MFD is left coprime, B,, = 0 and rank [Dlo BIoI] = 2 = p, then 836
Matrix Fraction Descriptions rank [E B] = 4 = n. Hence the above realized system is controllable infinite modes.
at finite and
Example 6 Consider
left MFD as follows :
a nonstrictly-proper
-2-l
s2+1
T(s) =
-s3_22s2-_s_2
[
,&$_s
-3s2-2s-2 =
-1
0 -3s2-2s-2
0
L =
_I
-3s2-2s-2
I
s2+1 L
-s2-
_s3-2s2_s_2
1
s3_s2_s
I
D; ’ (s)N,(s).
From the given MFD structure
&,=[-;
Since H, =
we have
;],
1
&=[-;
-;
0
o , we hence choose
corresponding
_;
_;I
-l/3
1
o [
_;
0
[
state-space
representation
1
1 , and the
o
Q0 = I2 and P, = form is y(t) = Cx(t),
I%(t) = Ax(t) +B,u(t)+B,ti(t), where -1 0
0
0
0
0
1
0
0
0
,A=
E= 0
0
0
1
0
0
0
0
0
0
-
1
-1
0
B,=
-213
0
0
O-
1
-213
0
0
0
0
0
0
0
2
0
0
1
0
2
0
0
1
3_
0
o-
‘0
-0 -
1
0
-2
0
-1
-1
-2
-1
,B,= -I
-1
0
0
0
0 1
and voi.
330, No. 5, pp. Xl5Ks39, 1993 Printed in Great Britain
837
W. S. Chen and J. S. H. Tsai 0 c=o
[
1
-113
0
0
0
0
0
0
1’
As the given MFD is left coprime and rank [& B,,I-~Dll,PoQoBll] = 2; i.e. rank [E B] = 5 = n, so the realized system is controllable at finite and infinite modes. Obviously, for the given coprime MFD, the minimal realization in state-space representation form can be obtained in this paper, even though [D,(s) N,(s)] is not row reduced. Note that the realization algorithm proposed in (8) does not work for this nonstrictly proper MFD.
v. Conclusions In this paper, we propose generalized, feasible and constructive algorithms to realize the column-pseudoproper(column-proper), row-pseudoproper(rowproper) and nonstrictly proper MFDs in the state-space representations with the quasi-controller(controller), the quasi-observer(observer) canonical forms and the system in (2), respectively. If the’ MFD is right(left) coprime, then the algorithm provides a minimal realization in state-space representation (i.e. the realized system is controllable and observable). Furthermore, if the right(left) MFD is column(row)-pseudoproper or nonstrictly-proper, then the realized system is controllable(observable) in the sense of Rosenbrock and Cobb, and we can use a simple method to check the observability(controllability) at infinity in this paper. Therefore, it appears ours is a simpler and more natural generalization than existing algorithms, thus we can regard these as a more general bridge of the conceptual gap between regular and singular systems.
References (1) D. G. Luenberger, “Dynamic equations Control, Vol. 22, pp. 312-321, 1977.
in descriptor form”, IEEE
Trans. Autom.
(2) G. C. Verghese, B. C. Leuy and T. Kailath, “A generalized state-space for singular system”, IEEE Trans. Autom. Control, Vol. 26, pp. 81 I-831, 1977. (3) T. F. S. Mohamed, “A new approach for designing a reduced-order controller of a linear singular system”, IEEE Trans. Autom. Control, Vol. 35, pp. 492495, 1990. (4) W. A. Wolovich, “The determination of state-space representations for linear multivariable systems”, IEEE Trans. Autom. Control, Vol. 9, pp. 97-106, 1973. (5) B. J. Bender and A. J. Laub, “The linear-quadratic optimal regulator for descriptor systems”, IEEE Trans. Autom. Control, Vol. 32, pp. 672-687, 1987. (6) J. Y. Lin and Z. H. Yang, “Optimal control problems for singular systems”, Znt. J. Control, Vol. 47, pp. 1915-1924, 1988. (7) Y. Y. Wang, S. J. Shi and Z. J. Zhang, “A descriptor-system approach to singular perturbation of linear regulators”, IEEE Trans. Autom. Control, Vol. 33, pp. 370373, 1988. (8) J. S. H. Tsai and W. S. Chen, “Generalized realization with a special coordinate for matrix fraction description”, Int. J. Systems Sci., Vol. 23, pp. 2197-2217, 1992. (9) S. Tan and J. Vandewalle, “Realization algorithm for determining generalized statespace representations”, Int. J. Control, Vol. 45, pp. 113771146, 1987.
838
Journal of the Franklin lnstltute Pergamon Press Ltd
Matrix
Fraction Descriptions
(10) S. Tan and J. Vandewalle, “A singular system realization for arbitrary matrix fraction descriptions”, IEEE Int. Symp. Circuit Systems, pp. 615-618, 1988. (11) T. Shiotsuki and S. Kawaji, “On a canonical form of descriptor systems”, Proc. 27th Conf. Deci. Control, pp. 2089-2090, 1988. (12) C. H. Fang and F. R. Chang, “Realization algorithm for constructing a controllable representation of a singular system with a special coordinate”, ht. J. Control, Vol. 50, pp. 1217-1226, 1989. (13) C. H. Fang and F. R. Chang, “A strongly observable and controllable realization of descriptor systems”, Control Theory Adv. Technol., Vol. 6, pp. 133-141, 1990. (14) D. Cobb, “Controllability, observability and duality in singular systems”, IEEE Trans. Autom. Control, Vol. 29, pp. 1076-1082, 1984. (15) H. H. Rosenbrock, “Structural properties of linear dynamic systems”, ht. J. Control, Vol. 20, pp. 191-202, 1974. (16) L. Y. Dai, “Control problem for linear systems with input derivatives control”, Znt. J. Systems Sci., Vol. 19, pp. 1645-1653, 1988. (17) D. Chen and F. S. Feng, “Invariants, canonical forms, and minimal realizations for singular systems”, 10th IFAC World Congress, Munchen, Vol. 9, pp. 121, 1987. (18) H. H. Rosenbrock and A. C. Pugh, “Contributions to a hierarchical theory of systems”, Int. J. Control. Vol. 19, pp. 845-867. 1974. Received : 20 September 1992 Accepted : 30 March 1993
Vol. 330. No. 5, pp. 815-839, 1993 Printed in Great Britain
839