76
IEEE TlL4NSACTIONS O N AUTOMATIC CONTROL, FEBRUARY
Therefore, t.he permut.at,ion transformat.ion for both odd and even
matrix dimensions is
1973
The inner form of t.he above mat.& can be obtained as discussed
above by using t.he premultiplying matrix as fo1lon.s:
'.
~ I , ~ 2 1 ~ ~ 2 1 3 ~ ~ 3 W ~ ~ 3 2 1 1 5 ~ ~ 4 3 6 2 6 1' ~ ~ 4 5 3 6 9 . 1 1 , '
One can also obtain a general relationship for generating the permutat.ion mat.rix P for n-even or n-odd.
Similarly, one can obtain t,he inners from the minors as follows:
I
=
PTdfP.
(4)
The above is evident, from noting (1) and P T = P - l , where P-1 =
inverse P .
One caneasily verify that. thereexists another typeof permutation
transformation for (1) which can be expressed for both even and
odd "n," as follows:
P*,Pl2,P?3l1PWlP,
P34251,P3*2516
P5312~6,.
,
.. .
To obt,ain the corresponding innerwise mat.rix, for the Hurwitz
matrix, =e prendtiplythelatter
by the folloaing permutation
matrix.
For n-even [t,he permutationn1at.k dimension is (n - 1) X
The stabi1it.y condition is that [A:] be positive innemise (pi.)
plus condition 1) stated above.
For t.he same polynonlial, the st.ability condition within the unit
circle (11 is that
F(1) > 0,
F(-1)
>0
and that [A:] be positive innerwise.
To obtain,for inst.ance, [Ai], we have
[A;]
= .X3
+ I; =
Similarly, [ A , ] can be ~ b t a i n e d . ~
The innerwise matrix [A;] can be transformeddirectly
into a
minor form, whereby the inner [A:] becomes the principal minor
and the determinant, values of the mat.rices are as follows:
CONCLUSION
I n comparing t.he inner form for the left-half plane and for the unit
circle, the first element of t,he second row in both cases is zero. This
is the unifying feature. The minor form has no suchunifying pattern.
Furt,hermore, there exist6 no inner-minor transformation thatmakes
t.he first. entry of the third row of [a,'] zero.
REFERENCES
The innerwise matrk has the unified form of left, of triangls of
zeros, as shown in earlier publications [ 11, [3], [4].
Finally, t o show that for the general problem of root. clustering
the innerwise mat,rirls offers a. unifJ4ng form which has a pattern
t.hat can be utilized advantageously for computational purposes, the
following example is discussed.
[ l ] E. I. Jury. 'Inners'approachtosomeproblems
of s p t e m theory," I E E E
Tron.s.Auioml. Cotltr..vol. AC16. pp.2337240, June
1971.
121 F. R. Gantrnacher. The T h e o r y of Matrrces. vol. 2. NewPork:Chelsea,
1959.
[3] E. I. Jury and S. 11.Ahn. ".I computational slgorit.hm for inners," I E E B
T r o n a . Automat. Contr. (Short Papers), vol. .4C-17, pp. 541-543, Aug. 1972.
141 --."Symmetric and
innerrrise matrices for the root-clustering and rootdistribution of apolynomial," J . Franklin Inst.. vol. 293, pp. 433-460,
June 19i2.
[ 6 ] B. D. 0. Andersonand E. I. Jury. "On asimplifiedSchur-Cohntest."
I E E E Trans. Automat. C o n t r . . to be published.
161 J. H. IVilkinson. The A l g e b r a i c Eigenralue P r o b l e m . New Tork: Oxford,
1965, pp. 44-45.
3 In a recent work by Anderson and JurJ- [ 5 ] .it is shown that [A:].
p i . can be
replaced b
:
-the simpler conditions on the linear combinations of the coefficients
of Fez).
ExabrpLs
Let
F(z)
= a.G*
+ u3z3 + a29 + a,z + ao,
a4 > 0.
The stabilitycondition in the open left-half plane is 1'21
1) a K > 0, k = 0,1,2,3,4;
2 ) in the following Hurwitz matrix,
2 Inthispattern.the
first sulwript in p , reiers
~
t o row location,andthe
second one refers to column location.
Comments on "On the Simplification of Linear Systems"
AI. I?. CHIDAAIBARA
The aut.hor of t.he above note' is to be complimented on presenting
a methodof approximating a higher order syst.em using the frequency
r-pome approach. It is interesting to note thst.t.here is good agreeManuscript received September 8. 1972.
of Science,
The author is XI-ith theSchool of Automation,IndianInstitute
Banglore 12, India.
1 T . C . Hsla, I E E E Trans. A d o m a t . Contr. (Tech.NotesandCorresp.),
vol. AC-li, pp. 372-374, June 1972.
77
TECHNICAL NOTES AS33 CORRESPONDENCE
TABLE I
Unit Step
Response
Uodel
Nz
1
Remarks
-
1 - (6/5)e-f
(1/5)e+’
?I3
lo-*
3.3 X
(7/6)e-t
+(1/6)edzt
x
+
1 - 1.2308e-1.043t
+ 0.2308e-z.4i
2.38
1.2 X
Second-order
model with t.wo
poles unspecified
1 - 1.Xe-1.043t
0.24e-2.4t
0.6188 X
Poles obtained
in
N 3 assigned to
the secondorder model
+
x 4
(C-II)r
1 - e--0.914t
+
0.967
Original 1 - e - t - e--2i
system
2e-St - e-4t
(fourt,h
order )
+
x
TABLE I1
10-4
1 - 1.21695e-1~0299t0.415 x
0.21695e-2~4001t
1.05 X 10-3
1 - e-0.923i
(C-III3
Predominant poles
-1, -2 retained in
the
model
Model
x 3
0.2917s
1
0 . 3 9 9 ~ ~ 1.375s
(C-IIk
+ 2)
+
+1
+
+
+I
0.29583s + 1
0.40458~~
+ 1.387537s + 1
+1
+1
1
1.09409s
ment. bekeen thet,ransient. response of the higher order syst.em and
that of the corresponding lower order model, which is derived by
mat,ching t h e m q i t u d e s of the frequency responses. However, a
few points discussed in the abovenote’ need some clarification.
Strictly speaking, the simplzed models developed by Davison
[l], [5], viz., D m and D,, andbyChidambara
[ 2 ] - [ 4 ] , namely,
C1 and 62, should not be used for comparison with the model presented by Hsia, because the basis on which t.hese different models
are developed is entirely different from that on which the model is
derived by Hsia.
If a comparison of the int.egral of the squarederror is valid at all,
reference should bemadeto
t.he models derivedusingt.hetwo
techniques described in [6] in view of which the models C1 and C2
become obsolete.
Denotethe lower order model derivedfrom Technique I [6],
which retainsthe
predominant, eigenvalues, by C-I, andthat
derived from the other met.hod [6] by C-11. It is a0rt.h recalling
[6] t.hat. Technique I1 allows one t,o assign any desired eigenvalues
to the lower order model. Let N refer to the model derived by
method proposed by Hsia.’
Table I gives a comparison of t.he quality of the different simplified
models for the same fourt,h-order example‘ [ 4 ] . Table I1 shows a
comparison of thetransfer functions of the various loner order
models.
Table I1 demonstrates that the simplified models based on Hsia’s
frequency response approach and Chidambara’s transient response
method’ [6] have approximately t,he same transfer function. Does
this imply that there is a one-to-one (at least in an approximate
sense) correspondence between the approximations based on frequency response approach and the transicntresponse approach?
REFERENCES
Tram. Automai. Cotltr., vol. AC-11. pp. 93-101, Jan. 1966.
+
1
1.0833s
Exact system
M . R . Chidambara, “On ‘A met.hodforsimplifyinglineardynamic
+
0.26641s
1
0 . 3 9 9 5 5 ~ ~ 1.37559s
Both poles unspecified
Firstrorder model
E. J. Davison. “ 4 method for simplifying linear dynamic systems.”
+2
x 2
10-3
0.0
Transfer Function
0.8333s
(s
l)(s
IEEB
sys-
cems.:::IEEE Trans. Automat. Contr. (Corresp.). vol. AC-12. pp. 119-120,
t e b . lY01.
E. J . Davison, “Aut.hor’s reply,” I E E E Trans. Automat. Contr. (Corresp.).
vol. AC-12, pp. 120-12f: Feb. 1967.
Further
remarks on simpliiying
linear
dynamic
131 PI. R . Chidambara.
systems,” I E B E Trans.Automat.Contr.
(Corresp.). vol. AC-12,pp. 213214, Apr. 1967.
E. J. Davison“Author’sreply,” I E E E T r a n s . A u t o m i . Contr. (Corresp.),
vol. AC-12. p.’214, ,%x.1967.
[4 I &I. R . Chidambara. Further comments on ‘ A method for simplifying linear
dynamic systems.’ ” IEEE Trans. Automat. Contr. (CorresD.).
. . vol. AC-12.
p: i99, Dee. 1967;
E. J. Danson,Reply
by E. J. Davison,” IEEE Trans.Automat. Confr.
(Corresp.), vol. AC-12. p. 799, Dee. 196i.
M.
R . Chidambara“Furthercomments
by M. R. Chidambara,” TREE
Trans. Automat. C o k r . (Corresp.), vol. AC-12,. pp. 799-800, Dee. 1967.
E. J. Davison, “Further reply by
E. J. Damson,” IEEE Trans. Automat.
Contr. (Corresp.). vol. AC-12. p. 800, Dec. 1967.
new method for simplifying large linear dynamic systems,” IEEE
[5] --,“A
Trans. Automat. Cord;. (Corresp.). rol. AC-13. pp. 214-215, Apr. 1968.
I61 kf. R . Chidambara, Two simple techniques ior thesimplification of large
dynamicsystems,”in
1969 Joint Automatic Control Conj., Preprints, pp.
669-674.
“A methodforlinearsystemsimplification,”in
Proc. l S t h
(71 T. C.Hsia,
Mlidaest Symq; Circuit Theory, 1970.
181 S.-M. Wang.Compensationof,?continuousautomaticcontrolsystemby
Automat. Remote Contr. (USSR), 1.01. 20,
means of adelayelement.filter,
pp. 421433. 1959.
Author’s Reply2
T. C. HSIA
I wish to thank Prof. Chidambara for his interest in my note.
I amvery grateful for his est.ablishing more comparisons of my
met.hod t.o the best of his met.hods. These results together with those
in my n0t.e’ should provide the reader with a good view of the performances of the wide-range of techniques for system simplification.
I would like, however, to make the following remarks in response to
Prof. Chidambara’s comments.
1) The intentof my comparison study was to demonstratet.hat the
new method is valid and useful. The models C1, C?,D,,and D , were
chosen in t.he study simply because t,hey are readily available and
are well knom-n to the readers. The use of the integral-square-error
criterion seem5 to me to be the mast common, as well unbiased, in
that none of these models was in any way related t.0 this criterion.
2) On the ot,her hand, the models C-I and C-I1 of Chidambara
are indeed derived on the basis of minimizat,ion of the step response
my frequency
integral-square-error. So to compare themwith
domain method is rather inappropriate. Even then, the
IYmodels
compare rather well. This result hasfurtherdemonstratedthe
good quality of my method.
3) It is noted that t,he approximation accuracy of the X models can
be further improved if more thau ( p q ) equations in ( 9 )of my not,el
are matched.
+
2 Manuscript received Sept.ember22, 1972.
The author is with %he Department of Electrical Engineering, Univer3it.y
Caliiornia, Davis, Calif. 95616.
of