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Irreducible realization in canonical forms
IrreducibleRealization in Canonical Forms by K. B. DATTA Department of Applied Physics University College of Technology, 92, Acharya P.C. Road, Calcutta-g.
India
This paper describes a computational method for the realization of a finite dimensional, time invariant dynamical system whose description is available in terms of a transfer m&ix or input-output data. The realization is irreducible and relies for its computation on the properties of a Hankel matrix, and obtains a state variable description of the system in either controllable or observable canonical forms.
ABSTRACT:
I.
Introduction
The problem of realizing a state variable model of minimal dimension for a time invariant, finite dimensional dynamical system when its description is available in terms of a transfer matrix, input-output data or impulse-response matrix, are well investigated in the literature (1-13). Basically, there are two approaches to the problem: one which uses the properties of polynomial matrices, and the other preferred by researchers which formulates and solves the problem via the Hankel matrix and its properties. Few workers (13) have devoted themselves to obtaining the irreducible realization directly in state variable canonical forms with which it is easy to study the performance characteristics of the system because these forms generally have a minimum number of parameters (14). The realization in (13) is not in controllable or observable canonical forms [see (12,15,16)] which are well adapted to the study of linear multivariable systems. The aim of the present paper is to obtain an irreducible realization of a time invariant, finite dimensional dynamical system whose description is given in terms of a transfer matrix in either controllable or observable canonical forms. The method is based for its formulation and solution upon the Hankel matrix associated with the transfer function. The set of independent vectors of this matrix is employed to define another set of basis vectors which are used to describe a computational algorithm for its realization. II. System Description and Preliminaries Let a linear finite dimensional, time invariant, completely controllable and completely observable, dynamical system be represented by the state variable description: i(t)=Ax(t)+Bu(t) y(t)=
Cx(t)+Eu(t), 437
(2.1)
K. B. Dam
where x(t) is an n-state vector, y(t) is a p-output vector, u(t) an m-input vector, A, B, C and E are matrices of compatible dimensions. It is easy to see from (2.1) that the transfer function description of the same system is: T(s) = C(sl-A)-lB+E,
(2.2)
where T(s) is a matrix of order p X m and is a rational function of s. T(s) is called a proper transfer matrix if E is not a null matrix otherwise it is called strictly proper. If T(s) is given as a proper transfer matrix whose elements are rational function of s, the problem of realization is to find A, B, C and E. Definition 1. The realization (A, fi, c, E) of a rational transfer matrix T(s) is said to be in controllable canonical forms if
A=[A,],
where i, j=l,2,. .., having the form: 0 0
m
1 0
B=
,
; every element
0
***
0
1
***
0
Aii =
... 0 &i(l)
0
0
%C2)
&i(3)
A, of x is a matrix of order di X dj
19 Aii= [jl) .“,) ;ui;d
1
*-* *’
(2.3)
~=[c‘1,C;,...,~~],
*
&i(4)
if: j;
(2.3a,b)
every element Bi of fi is a matrix of order di X m having non-zero only in its last row where they begin from the i-th column:
elements
[0 .. .O, 1, bii+l,. . . , him],
,Ii= i=
1,2,. . . , m;
the forms of c and E are non-specific but C’i in c is a matrix of order having the elements: ill(i)
c&i)
***
cdi)
cdi)
- - -
ci =
Here di’s are the controllability 438
Q,(i) c&,(i)
1,2, . . . , m.
indices.
Journal of The Franklin
p x
di
(2.3~)
Institute
Irreducible Realization in Canonical Forms We shall be concerned here only with systems where in B the columns bi, M and in C the rows ci, i = 1,2,. . . , p are linearly independent. i=l,2,..., Definition 2. The realization (A, 8, C, E) of a rational transfer matrix T(s) is said to be in observable canonical forms if & B2
A=[A,],
B=
and
*
C = [Cr, C2, . . . , CJ,
(2.4)
[I6 p; every element
where i,j=l,2 ,..., having the form: 0 1 0
0 0 1
” * 0 “’ 0 “’ 0
.
.
...
.
0
0
*’ .
1
A, of A is a matrix of order CliX dr
b 0
@i(l) @i(2) &i(3) 7
A,=
0 . . .
”
0 . . .
’
‘*’
&j(l) &j(2) .
...
;
i# j;
(2.4a,b)
&($)
the form of B is non-specific, & x m having the elements
where every element
Bi is a matrix of order
h(i) 621(i) fji =
,
i=l,2
,...,
p;
(2.4~)
the element Cj of C is a matrix of order p x Ciihaving non-zero elements only in its last column where they begin from the i-th row: [0 * **0, 1,
Ci+l,iv. * CpilT*
The form of E is non-specific. C2i’sare here observability indices. To determine E given T(s) as a proper transfer matrix one has to take the limit of T(s) as s + m giving E = lim T(s). .9-C= Definition 3. The part left after subtracting Vol. 303, No. 5, May 1977
(2.5) E as obtained in (2.5) from the 439
K. B. Datta given T(s) is called reduced transfer matrix and is designated by G(s). Thus G(s) = T(s) - lim T(s) = C(sI- A)-‘B, J-r_
(2.6)
which is a strictly proper rational matrix as is easy to visualize. The problem of realization thus becomes to determine A, B, C from the reduced transfer matrix G(s). Expanding the right hand side of (2.6) about s = 0, we get G(s)
= f
His-(‘+‘),
where
Hi = CA’B.
(2.7)
i-0
If T(s) is given Hi’s are determined
by means of (2.6) and (2.7).
Definition 4. The triple (A, B, C) is a realization of the reduced transfer matrix G(s) if and only if CA’B = Hi, i = 0, 1, . . . , and is irreducible if and only if A, B is completely controllable and A, C is completely observable. By virtue of (2.7) a sequence of Hankel matrices Xba for all a, b = 0, 1, . . . is defined by Ho HI .* * II,_1 HI HZ *. . H,
(2.8)
The rank of %&, determines the order or degree of realization i.e. order of A in view of the following theorem proved in (7). Theorem I Let G(s) be a strictly proper rational matrix and let r be the degree of least common multiple of the denominators (Icd) of the elements in G(s), and let b and a be the first integers such that rank %$b = rank X,, = n. Then n is the degree of minimum realization of G(s), a is the observability index (max di) and b is controllability index (max di) of the minimal realization (A, B, C). Let us denote by hi(j) for i=O,l,..., r-l and j=I,2 ,.._, m the j-th column in the i-th column blocks headed by Hi in (2.8) when a = b = r. Then considering the matrix % = [ho(l), h&V,
. . . , ho(m); Ml), h(2), . . . , h(m); Ml), . . . , h-dm)l (2.9)
we can pick hi(j) if it is linearly independent of the vectors preceding it in (2.9) and group the vectors thus selected having the same value for j giving [h,(l), hi(l), . . . , h,,-I(l); ho@), . . . , h&2);
h#),
...,
k-l(m>l. (2.10)
The &‘s i=l,..., 440
m are called column indices of X’,,. Similarly considering Journal of The
Franklin Institute
Irreducible Realization rows we can determine row indices &‘s i = 1,. . . , can present our next theorem proved in (17).
in Canonical Forms
of %&.With these ideas we
Theorem II Let (A, B, C) be a realization of the reduced transfer matrix G(s) and 55%be the Hankel matrix associated with it where r is a degree of its ltd. Then column (row) indices of %$, are the same as the invariant controllability (observability) indices of the pair [A, B] ([A, C]) and vice uersa. As a consequence of the above two theorems we get: Theorem III If (A, B, C) is a realization of G(s) as given in (2.6) then rank of the matrix %, or 0, defined by %,=[B,AB
,...,
A’-‘B],
0, = [C’, A’C’, . . . , A”-‘C’]
(2.11)
where a prime denotes the transpose of a matrix and r is a degree of Icd of G(s) is n. III. Realization
in Controllable Canonical Forms
Let us consider the realization in controllable canonical forms. By the fact that (2.1) is completely controllable it is possible to construct a non-singular matrix Q (15)such that AQ = QA,
B = Q,,,&,,
(3.la,b)
where I& is a matrix of order m x n having i-th row equal to the ki( = CiE1 4)th iow of B in (2.3) i = 1,2,. . . , m and Q,,, is a matrix of order n X m whose i-th column is the same as the ki-th column in Q, i = 1,2, . . . , m. Since Bil is an upper triangular matrix having unit elements on the main diagonal, we have from (3.lb) denoting by Pki the remaining elements of Bil and by qi the i-th column of Q: i-l qkt =
h
+
C k=l
(3.2)
Pkibk.
Let p,,(i) = O,qh, then premultiplying both sides of (3.2) by 0, and observing from (2.8) and (2.9) that h,,(i) = Orbi for all i = 1,2,. . . , m we get i-l PO(i)
=
hOti>
+
k&
PkihO(k)*
(3.3)
By definition of column indices we can write da-1 h,(i)=
c j=O
m c k=l
i-l qkh#)+
2 k=l
wi,kh,(k).
Premultiplying both sides of (3.2) by 0+44 and comparing (3.4) it iS easily concluded that Pki = -(Y&k. Vol. 303. No. 5, May 1977
(3.4) the result with
441
K. B. Datta Definition 5. Let gj’S represent the columns of a matrix G and let gl(i) be the first non-zero element from the top in the i-th row of its first column. Adding suitable multiples of g, to the remaining columns the elements in their i-th rows can be reduced to zero. If gz(k) is the first non-zero element from the top in the k-th row of second column of G so reduced, then this column can be similarly employed to reduce to zero k-th row elements of all columns but for the first and the second. If any column has zero elements in course of computation one has to pass on to the next column. This process is continued till all columns are exhausted. These operations are called elementary column transformations. Elementary row transformations are defined similarly. Let W= K+l,d+l, d = max di) and a unit matrix of order (d + 1)~ be subjected to elementary column transformations of H whereby they are reduced to fi and T. Then it is a simple result of linear algebra that I-IT= fi. Here T is an upper triangular matrix having unit diagonal elements and is referred to a transforming matrix of H. Let be any of T. taking into (2.9) and we can /3ki=t,,,4++mdi+i;k=1,2
i-l;i=l,2
,...,
(3.5)
If define C$Ajq,, pi(i), then is evident (3.2), (3.3) Theorem III pi(i), i 1,2, ...,~TZ; j = 0, 1, . . . , di - 1 constitute a set of n linearly independent vectors. We now write (3.la) in terms of its columns for all i = 1,2, . . . , m giving us:
j=l
Aqki_1+2= qb_I+l + 2 aii(2)qk,y j-1
(3.6) &b-l=
qh-2
+
2 aji(di - l)qk,,
j=l &,
= qh-l+
2 aji(di)qk,* j=l
Substituting qki_l from the last equation into one preceding it and continuing this for q14_2,qki_3 in this way we get: s-1
qki_-s= A”qk -
,zo A’-‘-’
2 aji(di - r)qk,, S = 1, * * * 7 di - 1
(3.7a)
j=l
and A*‘&, = i A*~-’ 2 aji(d$ - r+ l)qk,. j=l r=l 442
(3.7b)
Journal of The Franklin Institute
Irreducible Realization in Canonical Forms
Premultiplying
both sides of (3.7b) by 0, there results pd,(i) = ? C aji(di r=l j=l
r+ l)p+-r(j),
(3.8)
where in view of (3.2) i-l
pj(i)=~i(i)+~~~8*iCll(k),
. ., m; j=O,
i=l,2,.
1,. ..,di.
(3.9)
Knowing Hi’s j=O, 1,. . . , one determines Pki'S with the aid of (3.5) and pj(i)‘s by (3.9) and constructs the following matrix: pd
=
pd+l,d+l
=[pO(l),
PO(~),
. . . , PO(~);
PI(~),
PI(~),
. . . , PI(m); pzw,
* * * , pdh)]
(3.10)
and subjects it to elementary COhmn transformations. Let Td be a transforming matrix of order (d + 1) m. Then in view of (3.8) (mdi + i)-th column of Td determines aii(d - r + 1) j = 1, . . . , m ; r = 1, . . . ,4. Again premultiplying both sides of (3.7b) by O,A” it follows that for s = 0, l,.. . . , d - di;
pd,+s(i)
=
?
2
apt4
-
r+
l)pd,--,+,(j).
(3.11)
r=lj=l
The above relation shows that for all s = 0, 1, . . . , d - di; i = 1,. . . , m, the {m(di + s) + i}-th columns in Td can be arranged to have the same elements in the rows ms+p, p=l,2 ,..., mdi+i. Let us now define: Pi = CA’[qk,, e. . , qk,], a matrix of order p x m i.e. _ _._ CA ‘B = PjB,, (3.12)
(3.13)
Aj =
forj=1,2,...,
(3.14)
d. Introducing
Vol.ao3,No.5,May1977
(3.12) into (3.10) we can write PdTd = $$id in the 443
K. B. Datta
form:
1;
1
... pd
pd+l
”
.
*
P2d
which is evident from (3.8) and (3.11) where in @d the submatrix constituted by the last m columns is a null matrix, Tii’s are submatrices of order m X m, I is a unit matrix of order m and Ai’s are defined in (3.14). Equating the (ki -s)-th columns on both sides of CO = C for all i = 1,2,. . . ) m we get
c[qkl-s,qkz-s, *. . , qk,,,-sl = cd-s,
(3.16)
where s = 0, 1,. . . , d - 1, 4i’S having non-positive subscripts are considered zero vectors. Substituting from (3.7a) and taking into account of (3.12) and (3.14) we get d-j-l Pd-j
-
1 i=O
PjAi+j+l = Cj,
j = 1, . ee7d.
(3.17)
Referring to (3.15) the above Eq. (3.17) conforms to the following representation by block triangular matrices:
x
cd-
0
Observe that in view of (3.2) BZ1 is an upper triangular matrix [pki] defined by (3.5) having unit diagonal elements. Theorem IV The triple (A, fi, C) is a realization of the reduced transfer matrix G(s) in controllable canonical form if and only if there exist matrices Aj’S (j = 1 9***9 d) of order m x m, Cj’S (j= 1,. . . , d) and Pi’s (j=O, 1,2,. . .) both of 444
Journal ofTheFranklin htirute
Irreducible Realization in Canonical Forms order p x m satisfying d-j-1
Pd-j-
PiAi+j+l=Cj,
C
i=O
j=1,2,...,d;
and d-l Pd+j
-
C
Pi+jAi+l
=0
for all
j” 0.
(3.19)
i=O
Proof: Given G_(s) and the realization (A, 8, c), the non-singular upper triangular matrix B, is defined with the non-zero rows of I3 as described in (2.3) and pi’s are determined as in (3.12). Defining Ai’s and Cj’s as in (3.13) and (3.14) the necessity of the theorem follows as a result of (3.15) and (3.18) and the discussion associated with them. The proof of sufficiency is similar to Theorem 4 in (17), only Pi’s in the present case must replace Hi’s in that theorem and hence omitted. Theorem V. The realization (A, 15, e) as defined by the controllable canonical form in Theorem IV is irreducible. Referring to Theorem I, it immediately follows from Theorem 6 in (7) that (A, fi, c) is an irreducible realization with ‘a’ as observability index and ‘b’ as controllability index. Theorem VI In the irreducible realization (A, B, c) in controllable canonical forms, as defined in Theorem IV, the non-fixed parameters describing A are minimum in number. Proof: To prove the Theorem, we observe as stated in (14) that the element Aji in the canonical form (2.3) is in terms of minimum number of parameters if its non-fixed parameters are aii(k), k = 1,2, . . . , min (d, dj). Assuming di > dj, we note from (3.9) that PO(j), . . . , pd,-1(j) are linearly independent and hence their coefficients +(l), . . . , aii(dj) in (3.8) are in general non-zero but those associated with the dependent vectors p4(j), . . . , p,+10’) aji(4 + 1)~ . . . 9 +(di) must be zero for all i, j = 1,2, . . . , m. This completes the proof. Example 1. Let
We note that here E = 0, T(s) = G(s) and r=4. Vol. 303. No. 5, May 1977
Constructing
Xd4 and its 445
%
o\
transforming matrix T, we get 0 0 1 1 -2 -3 3 ,7
0 1 1 -1 -3 0 7 4
1 1 -2 -3 3 7 -4 -15
1 -1 -3 0 7 4 -35 -16
-2 -3 3 7 -4 -15 5 31 %a4
-3 0 7 4 -15 -16 31 48
3 7 -4 -15 5 31 -6 -63
7 4 -15 -16 31 48 -63 -128
I 1
0
-1l-l
31
-121
021
2 0 0 0 0 0 1 5 0 1 0 0 4 -1 -2 0 2 -3 1 -1 3 1 -7
-1021
0 1 0 -2 -1 3 4 -3
1 0 0 1 0 -2 -2 2
0 0 0 0 0
0 0 0
T
0 0 0 0
-1 -1 4 5 %‘I
Hence dI = 3, dz = 1. Because m = 2, in view of (3.5), pIz = fS4= -1. By virtue of (3.9) we compute CFdin (3.10) and construct the first matrix in the left side of (3.15) and (3.18) which after elementary column transformation give us:
1
n w
0
41
0
11
r
,O
_-
:o ;u
u1
L_________
01
0 1 01000000 10000000 l-2 10 -2-l 0 -3 3 -2 3 4-2 -7-3 2
-0
=
n
”
I..
0 0
Oil 1;
0
^ u
_ 1 1
0:2 ^@a ’ Oil
0
0
0
0
0
0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
o-1 o-1 0 0
4 5
c,
____
0
c____-_ 1
0
___
_ c2 ___
0^ I ^ u_l Cl
Decentralized
Regulation using Local Controllers
and where
(1%
The remaining sub-matrices in Eq. (18) do not depend on Kij or on Gij and are given in the Appendix. It will now be demonstrated that the local controller gains K” and the local observer gains Gy that minimize (13) are given by K’(t)
= -R-lB’M(t)
Gy = Ppc’IRGf,
(20) i = 1,2
(21)
where -~;~‘:=@A+A’M-~&BR-‘B’@+Q M(T)
=0
PP=APP+PPA’-PPC,R~~~iPP+(~~~~)i
(23)
Pi(O)=E{X(O)X’(O)}.
To establish this result note that the evaluation of Eq. (13) for the closed-loop system (17), yields
where -ni,
= M,(A + BK) + (A + BK)‘k& + Q + K’RK
-iI&, =M,(-Bk)+M,,(& --tie = M&(-B@
with boundary
+ &(A
ec)+(A+BK)‘M,, - Gc) - l?B’M,.
(25) -K’RI?:
+ (A - @‘Me
(26) +&RI?
(27)
conditions, M,(T)=O,
M,,(T)=O,
M,(T)=O.
(28)
Since the positive semi-definite matrix Q appears only in (29, the optimum gain K is given by the same value as in the deterministic case, (20), with ti satisfying Eq. (22). Vol.. 303, No. 5, May 1917
477
K. B. Datta of the denominators of its elements and form & as defined in (2.8) by expanding it according to (2.7). (3) Subject %$, to elementary row (or column) operations as described in Definition 5 and find the observability (or controllability) indices $‘s i = II. Let T be the 19% * . . , p (or di’S, i = 1,2, . . . , m) following Theorem transforming matrix such that TH= fi (or HT= I;r> where H= %!d+l.d+l, d’= max di (or H= %!d+l,d+l, d = max di). Note that T and H are in general different in the process of obtaining observable and controllable forms, but for notational convenience they are shown to be the same. Further assume fiik=rti+i,d+k;k=1,2,...,i-i;
i=l,2,...,P,
(or Eq. (3.5)) and define linearly independent
(4.2)
II row (or column) vectors
i-l
fiik&(k), i=1,2,.
fii(i)~$(i)+~;~
. . ,P;j=O,
1,. . .,&,
(4.3)
where hi(i) denotes the i-th row of j-th row-block of H: see 2.8 (or define as (3.9)). (4) With the aid of vectors as defined in (4.3) for j now varying up to d’ (or in (3.9) for j varying up to d) a matrix 9% (or CZpd) is constructed and row (or column) transformed to @d (or gd) by the transforming matrix Td (or Td) such that Td $??J = $d (or gdTd = gd) is in the form: 7
‘Tll
. . 0 . Tal Td2--* T21
-
T22
=
CFd (4.4)
Tad
-A1
-A2
*..
-Ad
1
where in $d the submatrix constituted Ai’s are defined by
by the last p rows is a null matrix and
hll(&-i+i)
a12(&-d+i)
...
azl(ci;-d+i)
az2(d2-d+i)
...
Qpl(c?p-h+i)
&(;i,-6+i)
... i=
1,2,. . .,d,
(4.5)
Irreducible Realization in Canonical Forms in which the elements ati(k are identified with those given in (2.4a,b) (or (3.15) and (3.14) the elements aij(k)‘s are identified with those given in (2.3a,b)). (5) Having obtained the transforming matrix TJ (or Td) and knowing Pi’s h- 1 (or Pi’s i = 0, 1, . . . , d -1) as in (4.3) (or in (3.15)) we i=O, l,..., construct
-L T21
0 .
.
. . .
T’l
0 PO PO P,
T22
T&’ _-Al -A2
0
. .
0
-.
* .
0
=
*** T&j .*. -Ad I
*.
Bz
B1
(4.6) (or as given in (3.18)) where Bi’S (or C’s) are defined bd,-d+&l)
’ *’
b&-d+i,2(2)
**’
,
i=l,2,
...,d
(4.7)
(or as in (3.13)) in which b,(k)‘S (or cij(k)‘S) are identified with those given in (2.4~) (or (2.3~)). (6) Define fi = [&I, a lower triangular matrix (or /3 = [&I, an upper triangular matrix) by the elements as defined in (4.2) (or (3.5)) with unit diagonal elements. The i-th column (or row) of fi-’ (or p-‘) determines l;i=h,+. e.+&-th column (or ki = d,+. . .+di-th row) of B (or C).
Proof: The algorithm is a direct consequence IV, V, and VI. Example 2. Obtain an observable
--
of Theorem
canonical form of the transfer matrix: 5s2+10s+7 5s2+10s+ll
It is easy to see that here E =0, T(s)= G(s) and r=3. Vol. 303, No. 5. May 1977
VII (or Theorems
1 *
The matrix X3, is 449
6 0 -5 3 2 0 0
-4 0 7 0 0
11 7 7 -1) -14 -4 00 -28 -8 0 0
*O
I ji
0s
-30 00 1 -1.8 10 1 0 0:3 11 02 0 1 0 o;o 0:3 10 **1 1 0:1 1;o 1 A, , / A, : A3 :
I
-_
______________________________________
4
-
4
=
-5
-10
5
0
5
4 -5 -4 10
5 -10 -5
-0.
-5
-5 -4 5 11
4
-10
-10 -5 10 7 j
4
10
10 7 -10 -11
0~0000
5
; 5 : 11 : -5 i-25
5:-4 -5
5 -10 -5
5
-4
-5
-5 -25 5 46
-4 11 5
00 0
0
2
0
7 -1 -4
10 17 -10 -25 10
7 -11 -10
7 -14
pd
11
,5
5 46 -5 -74
11 -25 -5
-5
#‘a
-10
-10 -11 10 17
-5 10 7
0 0
-21 21
-25
-25 5 46 -5 -74 5 109 -5
24 0 0 0 0I
-3ti 6
-11
-11 10 17 -10 -25 10 35 -10 I
whence observability indices are dI = 3, & = 1. Because p = 2, in view of (4.2) p,, = td43 = -3. Then according to steps 4 and 5 of the Algorithm combining (4.4) and (4.6), we obtain
=
-4 5 (0 -3 0 0 0000 0 0 -0 0
constructed according to step 2 of the Algorithm and is subjected to elementary row transformations as follows:
’
Irreducible Realization in Canonical Forms We also observe that
Therefore,
irreducible
realization
in observable
canonical form is:
It is interesting to observe that there are other realizations canonical forms of the transfer matrix in Example 2. viz.,
in observable
although not with minimum number of non-fixed parameters. V. Conclusion A computational algorithm is developed in this paper to obtain state variable description of a time invariant finite dimensional, completely controllable and completely observable canonical forms. Furthermore, the obtained canonical form of A will contain a minimum number of non-fixed parameters. The major disadvantage of the proposed method seems to be the evaluation of an inverse of an upper triangular matrix of order m or p which is not difficult in view of a known recursive algorithm (18). Whether the properties of polynomial matrices can be applied to obtain the canonical forms will be an interesting topic of future research. References (1)E. G. Gilbert, “Controllability (2) (3)
(4) (5)
and observability in multivariable control systems,” SIAM J. Control, Vol. 1, No. 2, pp. 128-151, 1963. R. E. Kalman, “Mathematical description of linear dynamical systems,” SIAM J. Control, Vol. 1, No. 2, pp. 152-192, 1963. B. L. Ho and R. E. Kalman, “Effective construction of linear state variable models from input-output functions”, Regelungstechnik, Vol. 14, No. 12, pp. 545-548, 1966. R. E. Kalman, “On structural properties of linear, constant, multivariable systems”, Preprint, 3rd. IFAC Cong. Session 6, paper 6.A, 1966. R.. E. Kalman, P. L. Falb and M. A. Arbib, “Topics in Mathematical System Theory Book”, McGraw-Hill, New York, pp. 237-339, 1969.
Vol. 303, No. 5, May 1977
451
K. B. Datta (6) R. E. Kalman, “Irreducible realizations and the degree of a rational matrix”, SIAM J. Appl. Math., Vol. 13, No. 2, pp.. 520-544, 1965. (7) L. M. Silverman, “Realization of linear dynamical systems”, Trans. IEEE on Automatic Control, Vol. AC-16, No. 6, pp. 554-567, 1971. (8) D. C. Youla and P. Tissi, “n-port synthesis via reactance extraction-Part l”, IEEE Int. Convention Rec., Vol. 14, Pt. 7, pp. 183-205, 1966. (9) D. Q. Mayne, “A computational procedure for the minimum realization of transfer function matrices”, Proc. IEE, Vol. 115, pp. 1363-1368, 1968. (10) P. E. Caines, “Minimum realization of transfer function matrices”, Control system research report, Centre for computing and Automation, March 1969. (11)C. Gori-Giorgi and A. Isidori, “A new algorithm for the irreducible realization of a rational matrix”, Recerche di Automatica, Vol. II, No. 3, pp. 225-293, 1971. (12) W. A. Wolovich, “Linear multivariable Systems”, Springer-Verlag, New York, pp. 77-93, 1974. (13) D. P. Mital and C. T. Chen, “Irreducible canonical form realization of a rational matrix”, Irat. J. Control, Vol. 18, No. 4, pp. 881-887, 1973. (14) M. J. Denham, “Canonical forms for multivariable systems”, IEEE Trans. Automatic Control, Vol. AC-19, No. 6, pp. 646-655, 1974. (15)P. Brunovskf, “On the stabilization of a linear system under a class of persistent perturbations”, Differential Equation (in Russian), Vol. II, No. 6, pp. 769-777, 1966. (16) P. Brunovsk$, “A classification bf linear controllable systems”,_ Kybernetika, Vol. 6, No. 3, pp. 173-188, 1970. (17) K. B. Datta, Canonical forms from input-output description, J. Inst. Electronics and Telecom. Engrs. (India). (18) K. B. Datta, Methods to compute canonical forms in multivariable control systems, .I. Inst. Electronics and Telecom. Engrs. (India).
452
Journal of The Franklin Institute
India
This paper describes a computational method for the realization of a finite dimensional, time invariant dynamical system whose description is available in terms of a transfer m&ix or input-output data. The realization is irreducible and relies for its computation on the properties of a Hankel matrix, and obtains a state variable description of the system in either controllable or observable canonical forms.
ABSTRACT:
I.
Introduction
The problem of realizing a state variable model of minimal dimension for a time invariant, finite dimensional dynamical system when its description is available in terms of a transfer matrix, input-output data or impulse-response matrix, are well investigated in the literature (1-13). Basically, there are two approaches to the problem: one which uses the properties of polynomial matrices, and the other preferred by researchers which formulates and solves the problem via the Hankel matrix and its properties. Few workers (13) have devoted themselves to obtaining the irreducible realization directly in state variable canonical forms with which it is easy to study the performance characteristics of the system because these forms generally have a minimum number of parameters (14). The realization in (13) is not in controllable or observable canonical forms [see (12,15,16)] which are well adapted to the study of linear multivariable systems. The aim of the present paper is to obtain an irreducible realization of a time invariant, finite dimensional dynamical system whose description is given in terms of a transfer matrix in either controllable or observable canonical forms. The method is based for its formulation and solution upon the Hankel matrix associated with the transfer function. The set of independent vectors of this matrix is employed to define another set of basis vectors which are used to describe a computational algorithm for its realization. II. System Description and Preliminaries Let a linear finite dimensional, time invariant, completely controllable and completely observable, dynamical system be represented by the state variable description: i(t)=Ax(t)+Bu(t) y(t)=
Cx(t)+Eu(t), 437
(2.1)
K. B. Dam
where x(t) is an n-state vector, y(t) is a p-output vector, u(t) an m-input vector, A, B, C and E are matrices of compatible dimensions. It is easy to see from (2.1) that the transfer function description of the same system is: T(s) = C(sl-A)-lB+E,
(2.2)
where T(s) is a matrix of order p X m and is a rational function of s. T(s) is called a proper transfer matrix if E is not a null matrix otherwise it is called strictly proper. If T(s) is given as a proper transfer matrix whose elements are rational function of s, the problem of realization is to find A, B, C and E. Definition 1. The realization (A, fi, c, E) of a rational transfer matrix T(s) is said to be in controllable canonical forms if
A=[A,],
where i, j=l,2,. .., having the form: 0 0
m
1 0
B=
,
; every element
0
***
0
1
***
0
Aii =
... 0 &i(l)
0
0
%C2)
&i(3)
A, of x is a matrix of order di X dj
19 Aii= [jl) .“,) ;ui;d
1
*-* *’
(2.3)
~=[c‘1,C;,...,~~],
*
&i(4)
if: j;
(2.3a,b)
every element Bi of fi is a matrix of order di X m having non-zero only in its last row where they begin from the i-th column:
elements
[0 .. .O, 1, bii+l,. . . , him],
,Ii= i=
1,2,. . . , m;
the forms of c and E are non-specific but C’i in c is a matrix of order having the elements: ill(i)
c&i)
***
cdi)
cdi)
- - -
ci =
Here di’s are the controllability 438
Q,(i) c&,(i)
1,2, . . . , m.
indices.
Journal of The Franklin
p x
di
(2.3~)
Institute
Irreducible Realization in Canonical Forms We shall be concerned here only with systems where in B the columns bi, M and in C the rows ci, i = 1,2,. . . , p are linearly independent. i=l,2,..., Definition 2. The realization (A, 8, C, E) of a rational transfer matrix T(s) is said to be in observable canonical forms if & B2
A=[A,],
B=
and
*
C = [Cr, C2, . . . , CJ,
(2.4)
[I6 p; every element
where i,j=l,2 ,..., having the form: 0 1 0
0 0 1
” * 0 “’ 0 “’ 0
.
.
...
.
0
0
*’ .
1
A, of A is a matrix of order CliX dr
b 0
@i(l) @i(2) &i(3) 7
A,=
0 . . .
”
0 . . .
’
‘*’
&j(l) &j(2) .
...
;
i# j;
(2.4a,b)
&($)
the form of B is non-specific, & x m having the elements
where every element
Bi is a matrix of order
h(i) 621(i) fji =
,
i=l,2
,...,
p;
(2.4~)
the element Cj of C is a matrix of order p x Ciihaving non-zero elements only in its last column where they begin from the i-th row: [0 * **0, 1,
Ci+l,iv. * CpilT*
The form of E is non-specific. C2i’sare here observability indices. To determine E given T(s) as a proper transfer matrix one has to take the limit of T(s) as s + m giving E = lim T(s). .9-C= Definition 3. The part left after subtracting Vol. 303, No. 5, May 1977
(2.5) E as obtained in (2.5) from the 439
K. B. Datta given T(s) is called reduced transfer matrix and is designated by G(s). Thus G(s) = T(s) - lim T(s) = C(sI- A)-‘B, J-r_
(2.6)
which is a strictly proper rational matrix as is easy to visualize. The problem of realization thus becomes to determine A, B, C from the reduced transfer matrix G(s). Expanding the right hand side of (2.6) about s = 0, we get G(s)
= f
His-(‘+‘),
where
Hi = CA’B.
(2.7)
i-0
If T(s) is given Hi’s are determined
by means of (2.6) and (2.7).
Definition 4. The triple (A, B, C) is a realization of the reduced transfer matrix G(s) if and only if CA’B = Hi, i = 0, 1, . . . , and is irreducible if and only if A, B is completely controllable and A, C is completely observable. By virtue of (2.7) a sequence of Hankel matrices Xba for all a, b = 0, 1, . . . is defined by Ho HI .* * II,_1 HI HZ *. . H,
(2.8)
The rank of %&, determines the order or degree of realization i.e. order of A in view of the following theorem proved in (7). Theorem I Let G(s) be a strictly proper rational matrix and let r be the degree of least common multiple of the denominators (Icd) of the elements in G(s), and let b and a be the first integers such that rank %$b = rank X,, = n. Then n is the degree of minimum realization of G(s), a is the observability index (max di) and b is controllability index (max di) of the minimal realization (A, B, C). Let us denote by hi(j) for i=O,l,..., r-l and j=I,2 ,.._, m the j-th column in the i-th column blocks headed by Hi in (2.8) when a = b = r. Then considering the matrix % = [ho(l), h&V,
. . . , ho(m); Ml), h(2), . . . , h(m); Ml), . . . , h-dm)l (2.9)
we can pick hi(j) if it is linearly independent of the vectors preceding it in (2.9) and group the vectors thus selected having the same value for j giving [h,(l), hi(l), . . . , h,,-I(l); ho@), . . . , h&2);
h#),
...,
k-l(m>l. (2.10)
The &‘s i=l,..., 440
m are called column indices of X’,,. Similarly considering Journal of The
Franklin Institute
Irreducible Realization rows we can determine row indices &‘s i = 1,. . . , can present our next theorem proved in (17).
in Canonical Forms
of %&.With these ideas we
Theorem II Let (A, B, C) be a realization of the reduced transfer matrix G(s) and 55%be the Hankel matrix associated with it where r is a degree of its ltd. Then column (row) indices of %$, are the same as the invariant controllability (observability) indices of the pair [A, B] ([A, C]) and vice uersa. As a consequence of the above two theorems we get: Theorem III If (A, B, C) is a realization of G(s) as given in (2.6) then rank of the matrix %, or 0, defined by %,=[B,AB
,...,
A’-‘B],
0, = [C’, A’C’, . . . , A”-‘C’]
(2.11)
where a prime denotes the transpose of a matrix and r is a degree of Icd of G(s) is n. III. Realization
in Controllable Canonical Forms
Let us consider the realization in controllable canonical forms. By the fact that (2.1) is completely controllable it is possible to construct a non-singular matrix Q (15)such that AQ = QA,
B = Q,,,&,,
(3.la,b)
where I& is a matrix of order m x n having i-th row equal to the ki( = CiE1 4)th iow of B in (2.3) i = 1,2,. . . , m and Q,,, is a matrix of order n X m whose i-th column is the same as the ki-th column in Q, i = 1,2, . . . , m. Since Bil is an upper triangular matrix having unit elements on the main diagonal, we have from (3.lb) denoting by Pki the remaining elements of Bil and by qi the i-th column of Q: i-l qkt =
h
+
C k=l
(3.2)
Pkibk.
Let p,,(i) = O,qh, then premultiplying both sides of (3.2) by 0, and observing from (2.8) and (2.9) that h,,(i) = Orbi for all i = 1,2,. . . , m we get i-l PO(i)
=
hOti>
+
k&
PkihO(k)*
(3.3)
By definition of column indices we can write da-1 h,(i)=
c j=O
m c k=l
i-l qkh#)+
2 k=l
wi,kh,(k).
Premultiplying both sides of (3.2) by 0+44 and comparing (3.4) it iS easily concluded that Pki = -(Y&k. Vol. 303. No. 5, May 1977
(3.4) the result with
441
K. B. Datta Definition 5. Let gj’S represent the columns of a matrix G and let gl(i) be the first non-zero element from the top in the i-th row of its first column. Adding suitable multiples of g, to the remaining columns the elements in their i-th rows can be reduced to zero. If gz(k) is the first non-zero element from the top in the k-th row of second column of G so reduced, then this column can be similarly employed to reduce to zero k-th row elements of all columns but for the first and the second. If any column has zero elements in course of computation one has to pass on to the next column. This process is continued till all columns are exhausted. These operations are called elementary column transformations. Elementary row transformations are defined similarly. Let W= K+l,d+l, d = max di) and a unit matrix of order (d + 1)~ be subjected to elementary column transformations of H whereby they are reduced to fi and T. Then it is a simple result of linear algebra that I-IT= fi. Here T is an upper triangular matrix having unit diagonal elements and is referred to a transforming matrix of H. Let be any of T. taking into (2.9) and we can /3ki=t,,,4++mdi+i;k=1,2
i-l;i=l,2
,...,
(3.5)
If define C$Ajq,, pi(i), then is evident (3.2), (3.3) Theorem III pi(i), i 1,2, ...,~TZ; j = 0, 1, . . . , di - 1 constitute a set of n linearly independent vectors. We now write (3.la) in terms of its columns for all i = 1,2, . . . , m giving us:
j=l
Aqki_1+2= qb_I+l + 2 aii(2)qk,y j-1
(3.6) &b-l=
qh-2
+
2 aji(di - l)qk,,
j=l &,
= qh-l+
2 aji(di)qk,* j=l
Substituting qki_l from the last equation into one preceding it and continuing this for q14_2,qki_3 in this way we get: s-1
qki_-s= A”qk -
,zo A’-‘-’
2 aji(di - r)qk,, S = 1, * * * 7 di - 1
(3.7a)
j=l
and A*‘&, = i A*~-’ 2 aji(d$ - r+ l)qk,. j=l r=l 442
(3.7b)
Journal of The Franklin Institute
Irreducible Realization in Canonical Forms
Premultiplying
both sides of (3.7b) by 0, there results pd,(i) = ? C aji(di r=l j=l
r+ l)p+-r(j),
(3.8)
where in view of (3.2) i-l
pj(i)=~i(i)+~~~8*iCll(k),
. ., m; j=O,
i=l,2,.
1,. ..,di.
(3.9)
Knowing Hi’s j=O, 1,. . . , one determines Pki'S with the aid of (3.5) and pj(i)‘s by (3.9) and constructs the following matrix: pd
=
pd+l,d+l
=[pO(l),
PO(~),
. . . , PO(~);
PI(~),
PI(~),
. . . , PI(m); pzw,
* * * , pdh)]
(3.10)
and subjects it to elementary COhmn transformations. Let Td be a transforming matrix of order (d + 1) m. Then in view of (3.8) (mdi + i)-th column of Td determines aii(d - r + 1) j = 1, . . . , m ; r = 1, . . . ,4. Again premultiplying both sides of (3.7b) by O,A” it follows that for s = 0, l,.. . . , d - di;
pd,+s(i)
=
?
2
apt4
-
r+
l)pd,--,+,(j).
(3.11)
r=lj=l
The above relation shows that for all s = 0, 1, . . . , d - di; i = 1,. . . , m, the {m(di + s) + i}-th columns in Td can be arranged to have the same elements in the rows ms+p, p=l,2 ,..., mdi+i. Let us now define: Pi = CA’[qk,, e. . , qk,], a matrix of order p x m i.e. _ _._ CA ‘B = PjB,, (3.12)
(3.13)
Aj =
forj=1,2,...,
(3.14)
d. Introducing
Vol.ao3,No.5,May1977
(3.12) into (3.10) we can write PdTd = $$id in the 443
K. B. Datta
form:
1;
1
... pd
pd+l
”
.
*
P2d
which is evident from (3.8) and (3.11) where in @d the submatrix constituted by the last m columns is a null matrix, Tii’s are submatrices of order m X m, I is a unit matrix of order m and Ai’s are defined in (3.14). Equating the (ki -s)-th columns on both sides of CO = C for all i = 1,2,. . . ) m we get
c[qkl-s,qkz-s, *. . , qk,,,-sl = cd-s,
(3.16)
where s = 0, 1,. . . , d - 1, 4i’S having non-positive subscripts are considered zero vectors. Substituting from (3.7a) and taking into account of (3.12) and (3.14) we get d-j-l Pd-j
-
1 i=O
PjAi+j+l = Cj,
j = 1, . ee7d.
(3.17)
Referring to (3.15) the above Eq. (3.17) conforms to the following representation by block triangular matrices:
x
cd-
0
Observe that in view of (3.2) BZ1 is an upper triangular matrix [pki] defined by (3.5) having unit diagonal elements. Theorem IV The triple (A, fi, C) is a realization of the reduced transfer matrix G(s) in controllable canonical form if and only if there exist matrices Aj’S (j = 1 9***9 d) of order m x m, Cj’S (j= 1,. . . , d) and Pi’s (j=O, 1,2,. . .) both of 444
Journal ofTheFranklin htirute
Irreducible Realization in Canonical Forms order p x m satisfying d-j-1
Pd-j-
PiAi+j+l=Cj,
C
i=O
j=1,2,...,d;
and d-l Pd+j
-
C
Pi+jAi+l
=0
for all
j” 0.
(3.19)
i=O
Proof: Given G_(s) and the realization (A, 8, c), the non-singular upper triangular matrix B, is defined with the non-zero rows of I3 as described in (2.3) and pi’s are determined as in (3.12). Defining Ai’s and Cj’s as in (3.13) and (3.14) the necessity of the theorem follows as a result of (3.15) and (3.18) and the discussion associated with them. The proof of sufficiency is similar to Theorem 4 in (17), only Pi’s in the present case must replace Hi’s in that theorem and hence omitted. Theorem V. The realization (A, 15, e) as defined by the controllable canonical form in Theorem IV is irreducible. Referring to Theorem I, it immediately follows from Theorem 6 in (7) that (A, fi, c) is an irreducible realization with ‘a’ as observability index and ‘b’ as controllability index. Theorem VI In the irreducible realization (A, B, c) in controllable canonical forms, as defined in Theorem IV, the non-fixed parameters describing A are minimum in number. Proof: To prove the Theorem, we observe as stated in (14) that the element Aji in the canonical form (2.3) is in terms of minimum number of parameters if its non-fixed parameters are aii(k), k = 1,2, . . . , min (d, dj). Assuming di > dj, we note from (3.9) that PO(j), . . . , pd,-1(j) are linearly independent and hence their coefficients +(l), . . . , aii(dj) in (3.8) are in general non-zero but those associated with the dependent vectors p4(j), . . . , p,+10’) aji(4 + 1)~ . . . 9 +(di) must be zero for all i, j = 1,2, . . . , m. This completes the proof. Example 1. Let
We note that here E = 0, T(s) = G(s) and r=4. Vol. 303. No. 5, May 1977
Constructing
Xd4 and its 445
%
o\
transforming matrix T, we get 0 0 1 1 -2 -3 3 ,7
0 1 1 -1 -3 0 7 4
1 1 -2 -3 3 7 -4 -15
1 -1 -3 0 7 4 -35 -16
-2 -3 3 7 -4 -15 5 31 %a4
-3 0 7 4 -15 -16 31 48
3 7 -4 -15 5 31 -6 -63
7 4 -15 -16 31 48 -63 -128
I 1
0
-1l-l
31
-121
021
2 0 0 0 0 0 1 5 0 1 0 0 4 -1 -2 0 2 -3 1 -1 3 1 -7
-1021
0 1 0 -2 -1 3 4 -3
1 0 0 1 0 -2 -2 2
0 0 0 0 0
0 0 0
T
0 0 0 0
-1 -1 4 5 %‘I
Hence dI = 3, dz = 1. Because m = 2, in view of (3.5), pIz = fS4= -1. By virtue of (3.9) we compute CFdin (3.10) and construct the first matrix in the left side of (3.15) and (3.18) which after elementary column transformation give us:
1
n w
0
41
0
11
r
,O
_-
:o ;u
u1
L_________
01
0 1 01000000 10000000 l-2 10 -2-l 0 -3 3 -2 3 4-2 -7-3 2
-0
=
n
”
I..
0 0
Oil 1;
0
^ u
_ 1 1
0:2 ^@a ’ Oil
0
0
0
0
0
0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
o-1 o-1 0 0
4 5
c,
____
0
c____-_ 1
0
___
_ c2 ___
0^ I ^ u_l Cl
Decentralized
Regulation using Local Controllers
and where
(1%
The remaining sub-matrices in Eq. (18) do not depend on Kij or on Gij and are given in the Appendix. It will now be demonstrated that the local controller gains K” and the local observer gains Gy that minimize (13) are given by K’(t)
= -R-lB’M(t)
Gy = Ppc’IRGf,
(20) i = 1,2
(21)
where -~;~‘:=@A+A’M-~&BR-‘B’@+Q M(T)
=0
PP=APP+PPA’-PPC,R~~~iPP+(~~~~)i
(23)
Pi(O)=E{X(O)X’(O)}.
To establish this result note that the evaluation of Eq. (13) for the closed-loop system (17), yields
where -ni,
= M,(A + BK) + (A + BK)‘k& + Q + K’RK
-iI&, =M,(-Bk)+M,,(& --tie = M&(-B@
with boundary
+ &(A
ec)+(A+BK)‘M,, - Gc) - l?B’M,.
(25) -K’RI?:
+ (A - @‘Me
(26) +&RI?
(27)
conditions, M,(T)=O,
M,,(T)=O,
M,(T)=O.
(28)
Since the positive semi-definite matrix Q appears only in (29, the optimum gain K is given by the same value as in the deterministic case, (20), with ti satisfying Eq. (22). Vol.. 303, No. 5, May 1917
477
K. B. Datta of the denominators of its elements and form & as defined in (2.8) by expanding it according to (2.7). (3) Subject %$, to elementary row (or column) operations as described in Definition 5 and find the observability (or controllability) indices $‘s i = II. Let T be the 19% * . . , p (or di’S, i = 1,2, . . . , m) following Theorem transforming matrix such that TH= fi (or HT= I;r> where H= %!d+l.d+l, d’= max di (or H= %!d+l,d+l, d = max di). Note that T and H are in general different in the process of obtaining observable and controllable forms, but for notational convenience they are shown to be the same. Further assume fiik=rti+i,d+k;k=1,2,...,i-i;
i=l,2,...,P,
(or Eq. (3.5)) and define linearly independent
(4.2)
II row (or column) vectors
i-l
fiik&(k), i=1,2,.
fii(i)~$(i)+~;~
. . ,P;j=O,
1,. . .,&,
(4.3)
where hi(i) denotes the i-th row of j-th row-block of H: see 2.8 (or define as (3.9)). (4) With the aid of vectors as defined in (4.3) for j now varying up to d’ (or in (3.9) for j varying up to d) a matrix 9% (or CZpd) is constructed and row (or column) transformed to @d (or gd) by the transforming matrix Td (or Td) such that Td $??J = $d (or gdTd = gd) is in the form: 7
‘Tll
. . 0 . Tal Td2--* T21
-
T22
=
CFd (4.4)
Tad
-A1
-A2
*..
-Ad
1
where in $d the submatrix constituted Ai’s are defined by
by the last p rows is a null matrix and
hll(&-i+i)
a12(&-d+i)
...
azl(ci;-d+i)
az2(d2-d+i)
...
Qpl(c?p-h+i)
&(;i,-6+i)
... i=
1,2,. . .,d,
(4.5)
Irreducible Realization in Canonical Forms in which the elements ati(k are identified with those given in (2.4a,b) (or (3.15) and (3.14) the elements aij(k)‘s are identified with those given in (2.3a,b)). (5) Having obtained the transforming matrix TJ (or Td) and knowing Pi’s h- 1 (or Pi’s i = 0, 1, . . . , d -1) as in (4.3) (or in (3.15)) we i=O, l,..., construct
-L T21
0 .
.
. . .
T’l
0 PO PO P,
T22
T&’ _-Al -A2
0
. .
0
-.
* .
0
=
*** T&j .*. -Ad I
*.
Bz
B1
(4.6) (or as given in (3.18)) where Bi’S (or C’s) are defined bd,-d+&l)
’ *’
b&-d+i,2(2)
**’
,
i=l,2,
...,d
(4.7)
(or as in (3.13)) in which b,(k)‘S (or cij(k)‘S) are identified with those given in (2.4~) (or (2.3~)). (6) Define fi = [&I, a lower triangular matrix (or /3 = [&I, an upper triangular matrix) by the elements as defined in (4.2) (or (3.5)) with unit diagonal elements. The i-th column (or row) of fi-’ (or p-‘) determines l;i=h,+. e.+&-th column (or ki = d,+. . .+di-th row) of B (or C).
Proof: The algorithm is a direct consequence IV, V, and VI. Example 2. Obtain an observable
--
of Theorem
canonical form of the transfer matrix: 5s2+10s+7 5s2+10s+ll
It is easy to see that here E =0, T(s)= G(s) and r=3. Vol. 303, No. 5. May 1977
VII (or Theorems
1 *
The matrix X3, is 449
6 0 -5 3 2 0 0
-4 0 7 0 0
11 7 7 -1) -14 -4 00 -28 -8 0 0
*O
I ji
0s
-30 00 1 -1.8 10 1 0 0:3 11 02 0 1 0 o;o 0:3 10 **1 1 0:1 1;o 1 A, , / A, : A3 :
I
-_
______________________________________
4
-
4
=
-5
-10
5
0
5
4 -5 -4 10
5 -10 -5
-0.
-5
-5 -4 5 11
4
-10
-10 -5 10 7 j
4
10
10 7 -10 -11
0~0000
5
; 5 : 11 : -5 i-25
5:-4 -5
5 -10 -5
5
-4
-5
-5 -25 5 46
-4 11 5
00 0
0
2
0
7 -1 -4
10 17 -10 -25 10
7 -11 -10
7 -14
pd
11
,5
5 46 -5 -74
11 -25 -5
-5
#‘a
-10
-10 -11 10 17
-5 10 7
0 0
-21 21
-25
-25 5 46 -5 -74 5 109 -5
24 0 0 0 0I
-3ti 6
-11
-11 10 17 -10 -25 10 35 -10 I
whence observability indices are dI = 3, & = 1. Because p = 2, in view of (4.2) p,, = td43 = -3. Then according to steps 4 and 5 of the Algorithm combining (4.4) and (4.6), we obtain
=
-4 5 (0 -3 0 0 0000 0 0 -0 0
constructed according to step 2 of the Algorithm and is subjected to elementary row transformations as follows:
’
Irreducible Realization in Canonical Forms We also observe that
Therefore,
irreducible
realization
in observable
canonical form is:
It is interesting to observe that there are other realizations canonical forms of the transfer matrix in Example 2. viz.,
in observable
although not with minimum number of non-fixed parameters. V. Conclusion A computational algorithm is developed in this paper to obtain state variable description of a time invariant finite dimensional, completely controllable and completely observable canonical forms. Furthermore, the obtained canonical form of A will contain a minimum number of non-fixed parameters. The major disadvantage of the proposed method seems to be the evaluation of an inverse of an upper triangular matrix of order m or p which is not difficult in view of a known recursive algorithm (18). Whether the properties of polynomial matrices can be applied to obtain the canonical forms will be an interesting topic of future research. References (1)E. G. Gilbert, “Controllability (2) (3)
(4) (5)
and observability in multivariable control systems,” SIAM J. Control, Vol. 1, No. 2, pp. 128-151, 1963. R. E. Kalman, “Mathematical description of linear dynamical systems,” SIAM J. Control, Vol. 1, No. 2, pp. 152-192, 1963. B. L. Ho and R. E. Kalman, “Effective construction of linear state variable models from input-output functions”, Regelungstechnik, Vol. 14, No. 12, pp. 545-548, 1966. R. E. Kalman, “On structural properties of linear, constant, multivariable systems”, Preprint, 3rd. IFAC Cong. Session 6, paper 6.A, 1966. R.. E. Kalman, P. L. Falb and M. A. Arbib, “Topics in Mathematical System Theory Book”, McGraw-Hill, New York, pp. 237-339, 1969.
Vol. 303, No. 5, May 1977
451
K. B. Datta (6) R. E. Kalman, “Irreducible realizations and the degree of a rational matrix”, SIAM J. Appl. Math., Vol. 13, No. 2, pp.. 520-544, 1965. (7) L. M. Silverman, “Realization of linear dynamical systems”, Trans. IEEE on Automatic Control, Vol. AC-16, No. 6, pp. 554-567, 1971. (8) D. C. Youla and P. Tissi, “n-port synthesis via reactance extraction-Part l”, IEEE Int. Convention Rec., Vol. 14, Pt. 7, pp. 183-205, 1966. (9) D. Q. Mayne, “A computational procedure for the minimum realization of transfer function matrices”, Proc. IEE, Vol. 115, pp. 1363-1368, 1968. (10) P. E. Caines, “Minimum realization of transfer function matrices”, Control system research report, Centre for computing and Automation, March 1969. (11)C. Gori-Giorgi and A. Isidori, “A new algorithm for the irreducible realization of a rational matrix”, Recerche di Automatica, Vol. II, No. 3, pp. 225-293, 1971. (12) W. A. Wolovich, “Linear multivariable Systems”, Springer-Verlag, New York, pp. 77-93, 1974. (13) D. P. Mital and C. T. Chen, “Irreducible canonical form realization of a rational matrix”, Irat. J. Control, Vol. 18, No. 4, pp. 881-887, 1973. (14) M. J. Denham, “Canonical forms for multivariable systems”, IEEE Trans. Automatic Control, Vol. AC-19, No. 6, pp. 646-655, 1974. (15)P. Brunovskf, “On the stabilization of a linear system under a class of persistent perturbations”, Differential Equation (in Russian), Vol. II, No. 6, pp. 769-777, 1966. (16) P. Brunovsk$, “A classification bf linear controllable systems”,_ Kybernetika, Vol. 6, No. 3, pp. 173-188, 1970. (17) K. B. Datta, Canonical forms from input-output description, J. Inst. Electronics and Telecom. Engrs. (India). (18) K. B. Datta, Methods to compute canonical forms in multivariable control systems, .I. Inst. Electronics and Telecom. Engrs. (India).
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Journal of The Franklin Institute